Authors Ivan Hu, Dieter van Melkebeek, Andrew Morgan,
License CC-BY-3.0
T HEORY OF C OMPUTING, Volume 20 (1), 2024, pp. 1–70
www.theoryofcomputing.org
Polynomial Identity Testing via
Evaluation of Rational Functions
Ivan Hu Dieter van Melkebeek Andrew Morgan
Received November 2, 2022; Revised April 7, 2024; Published July 18, 2024
Abstract. We introduce a hitting set generator for Polynomial Identity Testing based
on evaluations of low-degree univariate rational functions at abscissas associated
with the variables. We establish an equivalence up to rescaling with a generator
introduced by Shpilka and Volkovich, which has a similar structure but uses
multivariate polynomials.
We initiate a systematic analytic study of the power of hitting set generators by
characterizing their vanishing ideals, i. e., the sets of polynomials that they fail to hit.
We provide two such characterizations for our generator. First, we develop a small
collection of polynomials that jointly produce the vanishing ideal. As corollaries,
we obtain tight bounds on the minimum degree, sparseness, and partition class
size of set-multilinearity in the vanishing ideal. Second, inspired by a connection
to alternating algebra, we develop a structured deterministic membership test for
the multilinear part of the vanishing ideal. We present a derivation based on
alternating algebra along with the required background, as well as one in terms of
zero substitutions and partial derivatives, avoiding the need for alternating algebra.
A conference version of this paper appeared in the Proceedings of the 13th Innovations in Theoretical Computer
Science Conference [40].
ACM Classification: Theory of computation → Algebraic complexity theory,Theory of compu-
tation → Pseudorandomness and derandomization
AMS Classification: 68Q17, 68Q87, 68Q15
Key words and phrases: polynomial identity testing, derandomization, pseudorandomness,
lower bounds, vanishing ideal, Gröbner basis
© 2024 Ivan Hu, Dieter van Melkebeek, and Andrew Morgan
c b Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/toc.2024.v020a001
I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
As evidence of the utility of our analytic approach, we rederive known derandom-
ization results based on the generator by Shpilka and Volkovich and present a new
application in derandomization / lower bounds for read-once oblivious algebraic
branching programs.
1 Overview
Polynomial identity testing (PIT) is the fundamental problem of deciding whether a given
multivariate algebraic circuit formally computes the zero polynomial. PIT has a simple, efficient
randomized algorithm that only needs blackbox access to the circuit: Pick a random point and
check whether the circuit evaluates to zero on that particular point.
Despite the fundamental nature of PIT and the simplicity of the randomized algorithm, no
efficient deterministic algorithm is known—even in the white-box setting, where the algorithm
has access to the description of the circuit. The existence of such an algorithm would imply
long-sought circuit lower bounds [27, 2, 31]. Conversely, sufficiently strong circuit lower bounds
yield blackbox derandomization for all of BPP, the class of decision problems admitting efficient
randomized algorithms with bounded error [43, 28]. Although the known results leave gaps
between the two directions, they show that PIT constitutes an important stepping stone towards
derandomizing BPP, and suggest that derandomizing BPP can be achieved in a blackbox fashion
if at all.
Blackbox derandomization of PIT for a class of polynomials 𝒞 in the variables 𝑥 1 , . . . , 𝑥 𝑛 is
equivalent to the efficient construction of a substitution 𝐺 that replaces each 𝑥 𝑖 by a low-degree
polynomial in a small set of fresh variables such that, for every nonzero polynomial 𝑝 from 𝒞,
𝑝(𝐺) remains nonzero [49, Lemma 4.1]. We refer to 𝐺 as a generator, the fresh variables are its
seed, and we say that 𝐺 hits the class 𝒞. If there are 𝑙 seed variables, and if 𝑝 and 𝐺 have degree
at most 𝑛 𝑂(1) , then the resulting deterministic PIT algorithm for 𝒞 makes 𝑛 𝑂(𝑙) blackbox queries.
Much progress on derandomizing PIT has been obtained by designing such substitutions
and analyzing their hitting properties for interesting classes 𝒞. Shpilka and Volkovich [48]
introduced a generator, now dubbed the Shpilka–Volkovich generator, or “SV generator” for
short. It takes as an additional parameter a positive integer 𝑙 and can be viewed as an algebraic
version of 𝑙-wise independence in the sense that any selection of 𝑙 of the original variables
can remain independent while the others are forced to zero. The property is realized using
Lagrange interpolation with respect to 𝑛 distinct elements of the underlying field 𝔽 , one element
𝑎 𝑖 corresponding to each original variable 𝑥 𝑖 . We refer to the elements 𝑎 𝑖 as abscissas; they are
also parameters of SV.
Definition 1.1 (SV generator). The Shpilka–Volkovich (SV) Generator for 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] is paramet-
rized by the following data:
• A positive integer 𝑙.
• For each 𝑖 ∈ [𝑛], a distinct abscissa 𝑎 𝑖 ∈ 𝔽 .
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
The generator SV𝑙 takes as seed 𝑙 pairs of fresh variables (𝑦1 , 𝑧1 ), . . . , (𝑦 𝑙 , 𝑧 𝑙 ) and substitutes
𝑙
Õ
𝑥𝑖 ← 𝑧 𝑡 · 𝐿 𝑖 (𝑦𝑡 ), (1.1)
𝑡=1
where the Lagrange interpolant 𝐿 𝑖 is the unique univariate polynomial of degree at most 𝑛 − 1
satisfying 𝐿 𝑖 (𝑎 𝑖 ) = 1 and 𝐿 𝑖 (𝑎 𝑗 ) = 0 for 𝑗 ∈ [𝑛] \ {𝑖}.
SV1 takes two seed variables, 𝑦 and 𝑧. For any 𝑖 ∈ [𝑛], setting 𝑦 = 𝑎 𝑖 gets 𝑥 𝑖 = 𝑧 while the
other variables are set to zero. For larger 𝑙, SV𝑙 is the sum of 𝑙 independent copies of SV1 .
Shpilka and Volkovich proved that SV1 hits sums of a bounded number of read-once
formulas for 𝑙 = 𝑂(log 𝑛) [48], later improved to 𝑙 = 𝑂(1) [41]. The generator for 𝑙 = 𝑂(log 𝑛)
has also been shown to hit multilinear depth-4 circuits with bounded top fan-in [32], multilinear
bounded-read formulas [7], commutative read-once oblivious algebraic branching programs
[15], Σm ΣΠ𝑂(1) formulas (i.e., sums of terms that are the product of a monomial and a power
Ó
of a bounded-degree polynomial) [14], circuits with locally-low algebraic rank in the sense
of [37], and orbits of simple polynomial classes under invertible linear transformations of the
variables [39]. The generator is an ingredient in other hitting set constructions, as well, notably
constructions using the technique of low-support rank concentration [4, 3, 26, 25, 46, 10]. It also
forms the core of a “succinct” generator that hits a variety of classes, including depth-2 circuits
[19].
Vanishing ideal. In this paper, we initiate a systematic study of the power of a generator 𝐺
through the set of polynomials 𝑝 such that 𝑝(𝐺) vanishes, which we denote by Van[𝐺]. For any
fixed generator 𝐺, Van[𝐺] is closed under addition, and for all 𝑞 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] and 𝑝 ∈ Van[𝐺],
𝑞 · 𝑝 ∈ Van[𝐺]. By definition, this means that the set Van[𝐺] has the algebraic structure of an
ideal. From now on, we refer to Van[𝐺] as the vanishing ideal of 𝐺. Our technical contributions
can be understood as precisely characterizing the vanishing ideal of the SV generator.
Characterizations of the vanishing ideal facilitate two objectives:
Derandomization. A generator 𝐺 hits a class 𝒞 of polynomials if and only if 𝒞 and Van[𝐺]
have at most the zero polynomial in common. For a class 𝒞 defined by a resource bound,
𝐺 trivially hits 𝒞 if the characterization of the nonzero elements in Van[𝐺] is incompatible
with being computable within the resource bound. In other words, derandomization of
PIT for 𝒞 reduces to proving lower bounds for Van[𝐺]. By developing explicit structure
for polynomials in the ideal, lower bounds become more tractable.
More generally, given a characterization of Van[𝐺], in order to derandomize PIT for a class
𝒞 it suffices to design another generator 𝐺0 that hits merely the polynomials in 𝒞 ∩ Van[𝐺].
As 𝐺 hits the remainder of 𝒞, combining 𝐺 with 𝐺0 yields a generator for all of 𝒞. In this
way, one may assume—for free—additional structure about the polynomials in 𝒞, namely
that the polynomials moreover belong to Van[𝐺].
Lower bounds. If we happen to know that 𝐺 hits the class 𝒞 of polynomials computable
within some resource bound, then any expression for a nonzero polynomial in Van[𝐺]
T HEORY OF C OMPUTING, Volume 20 (1), 2024, pp. 1–70 3
I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
yields an explicit polynomial that falls outside 𝒞. Such a statement is often referred to
as hardness of representation, and it can be viewed as a lower bound in the model of
computation underlying 𝒞 (assuming the polynomial can be computed in the model at all).
Characterizing Van[𝐺] makes explicit the polynomials to which the lower bound applies.
We illustrate how to make progress on both objectives through our characterizations of the
SV generator’s vanishing ideal.
Rational function evaluations. Another contribution of our paper is the development of an
alternate view of the SV generator, namely as evaluations of univariate rational functions of
low degree. We would like to promote the perspective for its intrinsic appeal and applicability.
Among other benefits, it facilitates the study of the vanishing ideal.
The transition goes as follows. Recall in Definition 1.1 that the SV generator takes as
additional parameters a positive integer 𝑙 and an arbitrary choice of distinct abscissas 𝑎 𝑖 ∈ 𝔽 for
each of the original variables 𝑥 𝑖 , 𝑖 ∈ [𝑛]. When 𝑙 = 1, SV1 takes as seed two fresh variables, 𝑦
and 𝑧, and can be described succinctly in terms of the Lagrange interpolants 𝐿 𝑖 for the set of
abscissas. Plugging in an explicit expression for the Lagrange interpolants, we have:
Ö 𝑦 − 𝑎𝑗
𝑥 𝑖 ← 𝑧 · 𝐿 𝑖 (𝑦) 𝑧 · . (1.2)
𝑎𝑖 − 𝑎 𝑗
𝑗∈[𝑛]\{𝑖}
By rescaling, the denominators on the right-hand side of (1.2) can be cleared, resulting in the
following somewhat simpler substitution:
Ö
𝑥𝑖 ← 𝑧 · (𝑦 − 𝑎 𝑗 ). (1.3)
𝑗∈[𝑛]\{𝑖}
The vanishing ideals of (1.3) and SV1 are the same up to rescaling each variable to match the
rescaling from (1.2) to (1.3).
More importantly, we apply the change of variables 𝑧 ← 𝑧 0/ 𝑗∈[𝑛] (𝑦 − 𝑎 𝑗 ). The resulting
Î
substitution now uses rational functions of the seed:
𝑧0
𝑥𝑖 ← . (1.4)
𝑦 − 𝑎𝑖
The notion of vanishing ideal naturally extends to rational function substitutions. The change
of variables from (1.3) to (1.4) establishes that any polynomial vanishing on (1.3) also vanishes
on (1.4). The change of variables is invertible (the inverse is 𝑧 0 ← 𝑧 · 𝑗∈[𝑛] (𝑦 − 𝑎 𝑗 )), so any
Î
polynomial vanishing on (1.4) also vanishes on (1.3). We conclude that the vanishing ideal of
(1.4) is the same as that of SV1 up to rescaling the variables.
Note that, for fixed 𝑦 and 𝑧 0, (1.4) may be interpreted as first forming a univariate rational
𝑧0
function 𝑓 (𝛼) = 𝑦−𝛼 (depending on 𝑦 and 𝑧 0 but independent of 𝑖) and then substituting
𝑥 𝑖 ← 𝑓 (𝑎 𝑖 ). As 𝑦 and 𝑧 0 vary, 𝑓 ranges over all rational functions in 𝛼 with numerator degree
zero and denominator degree one. We denote (1.4) by RFE01 , where RFE is a short-hand for
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Rational Function Evaluation, 0 bounds the numerator degree, and 1 bounds the denominator
degree.
As a generator, RFE01 naturally generalizes to RFE𝑙𝑘 for arbitrary 𝑘, 𝑙 ∈ ℕ .
Definition 1.2 (RFE generator). The Rational Function Evaluation Generator (RFE) for 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ]
is parametrized by the following data:
• A non-negative integer 𝑘, the numerator degree.
• A non-negative integer 𝑙, the denominator degree.
• For each 𝑖 ∈ [𝑛], a distinct abscissa 𝑎 𝑖 ∈ 𝔽 .
The generator RFE𝑙𝑘 takes as seed a rational function 𝑓 ∈ 𝔽 (𝛼) such that 𝑓 can be written as
𝑔/ℎ for some 𝑔, ℎ ∈ 𝔽 [𝛼] with deg(𝑔) ≤ 𝑘, deg(ℎ) ≤ 𝑙, and ℎ(𝑎 𝑖 ) ≠ 0 for all 𝑖 ∈ [𝑛]. From 𝑓 , it
generates the substitution 𝑥 𝑖 ← 𝑓 (𝑎 𝑖 ) for each 𝑖 ∈ [𝑛].
There are multiple ways to parametrize the seed of RFE𝑙𝑘 using scalars; the flexibility to choose
is a source of convenience. We refer to Section 2 for a discussion on different parametrizations,
as well as on how large the underlying field 𝔽 must be. As is customary in the context of
blackbox derandomization of PIT, we assume that 𝔽 is sufficiently large, possibly by taking a
field extension.
The connection between RFE01 and SV1 extends as follows. For higher values of 𝑙, SV𝑙 is
defined as the sum of 𝑙 independent instantiations of SV1 . The same transformations as above
relate SV𝑙 and the sum of 𝑙 independent instantiations of RFE01 . Partial fraction decomposition
expresses a (non-degenerate) univariate rational function with numerator of degree 𝑙 − 1 and
denominator of degree 𝑙 as a sum of 𝑙 rational functions with numerators of degree 0 and
denominators of degree 1. As a result, SV𝑙 is equivalent in power to RFE𝑙−1 𝑙 , up to variable
rescaling. See Section 2 for a formal treatment.
For parameter values 𝑘 ≠ 𝑙 − 1, there is no SV generator that corresponds to RFE𝑙𝑘 , but
SV max(𝑘+1,𝑙)
encompasses RFE𝑙𝑘 (up to rescaling) and uses at most twice as many seed variables.
Thus, the RFE-generator and the SV-generator efficiently hit the same classes of polynomials.
However, RFE induces simple linear dependencies on the seed variables—as opposed to the
nonlinear dependencies produced by SV—which enables our approach for determining the
vanishing ideal. The moral is that, even though polynomial substitutions are sufficient for
derandomizing PIT, it nevertheless helps to consider rational substitutions. Their use may
simplify analysis and arguably yield more elegant constructions.
As another indication of the power of rational substitutions, an alternate interpretation of
the RFE generator is that it substitutes the ratio of two linear functions of the seed variables,
where the coefficients of the linear functions are powers of the abscissas. A generator that only
substitutes linear functions—as opposed to a ratio of linear functions—of the seed variables
must have seed length 𝑛 in order to hit all linear polynomials. This is because if the seed length
were less than 𝑛, then there exists a nontrivial linear combination of the 𝑛 variables that becomes
zero after substitution. In contrast, the simplest nontrivial case of RFE, RFE01 , hits all linear
polynomials and only needs a seed of length 2.
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
Generating set. Our first result describes a small and explicit generating set for the vanishing
ideal of RFE. It consists of instantiations of a single determinant expression.
Theorem 1.3 (generating set). Let 𝑘, 𝑙, 𝑛 ∈ ℕ and let 𝑎 𝑖 for 𝑖 ∈ [𝑛] be distinct elements of 𝔽 . The
vanishing ideal of RFE𝑙𝑘 in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] for the given choice of abscissas (𝑎 𝑖 )𝑖∈[𝑛] is generated by the
following polynomials over all choices of 𝑘 + 𝑙 + 2 indices 𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+2 ∈ [𝑛]:
h i 𝑘+𝑙+2
𝑘
EVC𝑙𝑘 [𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+2 ] det 𝑎 𝑖 𝑗 𝑎 𝑖𝑘−1
𝑗
... 1 𝑎 𝑖𝑙 𝑗 𝑥 𝑖 𝑗 𝑎 𝑖𝑙−1
𝑗
𝑥𝑖𝑗 ... 𝑥𝑖𝑗 . (1.5)
𝑗=1
Moreover, for any fixed set 𝐶 ⊆ [𝑛] of size 𝑘 + 1, the polynomials EVC𝑙𝑘 [𝐶 t 𝐿] form a generating set of
minimum size when 𝐿 ranges over all (𝑙 + 1)-subsets of [𝑛] that are disjoint from 𝐶, where
EVC𝑙𝑘 [𝑆] EVC𝑙𝑘 [𝑖1 , 𝑖2 , . . . , 𝑖 |𝑆| ]
for 𝑆 = {𝑖1 , . . . , 𝑖 |𝑆| } ⊆ [𝑛] with 𝑖1 < 𝑖2 < · · · < 𝑖 |𝑆| .
The name “EVC” is a shorthand for “Elementary Vandermonde Circulation”. Later in
this overview and in Section 9 we discuss a representation of polynomials using alternating
algebra, with connections to notions from network flow. In this representation, polynomials in
the vanishing ideal coincide with circulations, and instantiations of EVC are the elementary
circulations.
We refer to the set 𝐶 in Theorem 1.3 as a core. The core 𝐶 plays a similar role as in a
combinatorial sunflower except that, unlike the petals of a sunflower, the sets 𝐿 do not need to
be disjoint outside the core.
Example 1.4. Consider the special case where 𝑘 = 0 and 𝑙 = 1. The generator for Van[RFE01 ]
when 𝑖1 = 1, 𝑖2 = 2, and 𝑖3 = 3 is given by
1 𝑎1 𝑥1 𝑥1
EVC01 [1, 2, 3] 1 𝑎 2 𝑥 2 𝑥 2 = (𝑎 1 − 𝑎 2 )𝑥1 𝑥 2 + (𝑎 2 − 𝑎 3 )𝑥 2 𝑥 3 + (𝑎 3 − 𝑎 1 )𝑥 3 𝑥 1 .
1 𝑎3 𝑥3 𝑥3
For any fixed 𝑖 ∗ ∈ [𝑛], the polynomials EVC01 [𝑆] form a generating set of minimum size when 𝑆
ranges over all subsets of [𝑛] of size 3 that contain 𝐶 = {𝑖 ∗ }. As an aside, they also constitute
minimal polynomials not computable by read-once formulas, which is consistent with the fact
that SV1 hits all read-once formulas (see Theorem 5.7).
In general, the generators EVC𝑙𝑘 are nonzero, multilinear, homogeneous polynomials of
degree 𝑙 + 1, and they have nonzero coefficients for all multilinear monomials of degree 𝑙 + 1.
Each generating set of minimum size in Theorem 1.3 yields a Gröbner basis with respect to every
monomial order that prioritizes the variables outside 𝐶. A Gröbner basis is a special generating
set that allows solving ideal-membership queries more efficiently, among other problems in
computational algebra [12, 1]. Computing Gröbner bases for general ideals is exponential-space
complete [36, 38]. Theorem 1.3 represents a rare instance of a natural and interesting ideal for
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which we know a small and explicit Gröbner basis. See the end of Section 3 for more background
on Gröbner bases.
To gain some intuition about dependencies between the generators EVC𝑙𝑘 , note that permuting
the order of the variables used in the construction of EVC𝑙𝑘 yields the same polynomial or minus
that polynomial, depending on the sign of the permutation. This follows from the determinant
structure of EVC𝑙𝑘 and is the reason why we need to fix the order of the variables in order to
obtain a generating set of minimum size. More profoundly, the following relationship holds
for every choice of 𝑘 + 𝑙 + 3 indices 𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+3 ∈ [𝑛] and every univariate polynomial 𝑞 of
degree at most 𝑘:
h i 𝑘+𝑙+3
𝑘
det 𝑞(𝑎 𝑖 𝑗 ) 𝑎 𝑖 𝑗 𝑎 𝑖𝑘−1
𝑗
... 1 𝑎 𝑖𝑙 𝑗 𝑥 𝑖 𝑗 𝑎 𝑖𝑙−1
𝑗
𝑥𝑖𝑗 ... 𝑥𝑖𝑗 = 0. (1.6)
𝑗=1
The determinant in (1.6) vanishes because the first column of the matrix is a linear combination
of the next 𝑘 + 1. A minor expansion across the first column expresses the determinant of the
matrix as a linear combination of minors, and each minor is an instantiation of EVC𝑙𝑘 . Since (1.6)
vanishes, the minor expansion yields a linear dependency for every nonzero polynomial 𝑞 of
degree at most 𝑘. In fact, when {𝑖1 , . . . , 𝑖 𝑘+𝑙+3 } varies over subsets of [𝑛] containing a fixed core
of size 𝑘 + 1, the equations (1.6) generate all linear dependencies among instances of EVC𝑙𝑘 .
As corollaries to Theorem 1.3 we obtain the following tight bounds on Van[RFE𝑙𝑘 ]. The
bounds hold for every way to choose the parameters in Definition 1.2, including the abscissas.
Corollary 1.5. The minimum degree of a nonzero polynomial in Van[RFE𝑙𝑘 ] equals 𝑙 + 1.
Corollary 1.5 proves a conjecture by Fournier and Korwar [20] (additional partial results
reported in [35]) that there exists a polynomial of degree 𝑙 + 1 in 𝑛 = 2𝑙 + 1 variables that SV𝑙
fails to hit. The conjecture follows because the generators for Van[SV𝑙 ] have degree 𝑙 + 1 and
use 2𝑙 + 1 variables. See also Corollary 3.9 in Section 3.
As none of the generators contain a monomial of support 𝑙 or less, the same holds for
every nonzero polynomial in Van[RFE𝑙𝑘 ]. This extends the known property that SV𝑙 hits
every polynomial that contains a monomial of support 𝑙 or less [48]. See Proposition 5.1 and
Theorem 1.8 for a strengthening in the case of multilinear polynomials.
Corollary 1.6. The minimum sparseness, i. e., number of monomials, of a nonzero polynomial in
Van[RFE𝑙𝑘 ] equals 𝑘+𝑙+2
𝑙+1 .
The generators EVC𝑙𝑘 realize the bound in Corollary 1.6 as they exactly contain all multilinear
monomials of degree 𝑙 + 1 that can be formed out of their 𝑘 + 𝑙 + 2 variables. The claim that no
nonzero polynomial in Van[RFE𝑙𝑘 ] contains fewer than 𝑘+𝑙+2
𝑙+1 monomials requires an additional
combinatorial argument (see Lemma 6.1). It is a (tight) quantitative strengthening of the known
property that SV𝑙 hits every polynomial with fewer 𝑙
√ than 2 monomials [7, 26, 14, 19]. 𝑙Note that
𝑘+𝑙+2
for 𝑘 = 𝑙 − 1 we have that 𝑙+1 = 𝑙+1 = Θ(2 / 𝑙). One consequence is that for SV to hit all
2𝑙+1 2𝑙
polynomials with 𝑚 monomials, a seed length of 𝑙 = Ω(log 𝑚) is required. In particular, hitting
sparse polynomials requires 𝑙 = Ω(log 𝑛).
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Another consequence deals with set-multilinearity, a common restriction in works on
derandomizing PIT and algebraic circuit lower bounds. A polynomial 𝑝 of degree 𝑙 + 1
in a set of variables {𝑥 1 , . . . , 𝑥 𝑛 } is said to be set-multilinear if [𝑛] can be partitioned as
[𝑛] = 𝑋1 t 𝑋2 t · · · t 𝑋𝑙+1 such that every monomial in 𝑝 is a product 𝑥 𝑖1 · 𝑥 𝑖2 · · · · · 𝑥 𝑖 𝑙+1 , where
𝑖 𝑗 ∈ 𝑋 𝑗 . Note that set-multilinearity implies multilinearity but not the other way around.
As the generators EVC𝑙𝑘 are not set-multilinear, it is not immediately clear from Theorem 1.3
whether Van[RFE𝑙𝑘 ] contains nontrivial set-multilinear polynomials of any degree. However, a
variation on the construction of the generators EVC𝑙𝑘 yields explicit set-multilinear homogeneous
polynomials in Van[RFE𝑙𝑘 ] of degree 𝑙 + 1 where each 𝑋 𝑗 has size 𝑘 + 2 (see Definition 7.1). We
denote them by ESMVC𝑙𝑘 , where ESMVC stands for “Elementary Set-Multilinear Vandermonde
Circulation”. ESMVC𝑙𝑘 contains all monomials of the form 𝑥 𝑖1 · 𝑥 𝑖2 · · · · · 𝑥 𝑖 𝑙+1 with 𝑖 𝑗 ∈ 𝑋 𝑗 . For
any variable partition (𝑋1 , 𝑋2 , . . . , 𝑋𝑙+1 ) with |𝑋1 | = · · · = |𝑋𝑙+1 | = 𝑘 + 2, ESMVC𝑙𝑘 is the only
set-multilinear polynomial in Van[RFE𝑙𝑘 ] with that variable partition, up to a scalar multiple,
and exhibits the following extremal property. See also Theorem 7.4.
Corollary 1.7. The minimum partition class size of a nonzero set-multilinear polynomial of degree
𝑙 + 1 in Van[RFE𝑙𝑘 ] equals 𝑘 + 2.
Membership test. Our second characterization of the vanishing ideal of RFE can be viewed
as a structured membership test. Given a polynomial 𝑝, there is a generic way to test whether
𝑝 belongs to the vanishing ideal of a generator 𝐺, namely by symbolically substituting 𝐺 into
𝑝 and verifying that the result simplifies to zero. When 𝐺 is a polynomial substitution, the
well-known transformation of a generator into a deterministic blackbox PIT algorithm yields
another test: Verify 𝑝(𝐺) = 0 for a sufficiently large set of substitutions into the seed variables.
By clearing denominators, the same goes for rational substitutions like RFE𝑙𝑘 .
While the generic test works, one cannot extract 𝐺-specific insight into whether or why 𝐺 hits
any particular polynomial. In contrast, our membership test uses the specific structure of 𝐺 and
provides useful insight. Building on the generating set of Theorem 1.3, we state our structured
test for membership of multilinear polynomials in Van[RFE𝑙𝑘 ] in terms of partial derivatives
and zero substitutions. Several prior papers demonstrated the utility of those operations in the
context of derandomizing PIT using the SV generator, especially for syntactically multilinear
models [48, 32, 7].
Theorem 1.8 (membership test for multilinear polynomials). Let 𝑘, 𝑙, 𝑛 ∈ ℕ and let 𝑎 𝑖 for 𝑖 ∈ [𝑛]
be distinct elements of 𝔽 . A multilinear polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] belongs to Van[RFE𝑙𝑘 ] if and only
if both of the following conditions hold:
1. There are no monomials of degree 𝑙 or less, nor of degree 𝑛 − 𝑘 or more, in 𝑝.
2. For all disjoint subsets 𝐾, 𝐿 ⊆ [𝑛] with |𝐾| = 𝑘 and |𝐿| = 𝑙, 𝜕𝐿 𝑝| 𝐾←0 is zero upon the following
substitution for each 𝑖 ∈ [𝑛] \ (𝐾 ∪ 𝐿), where 𝑧 denotes a fresh variable:
𝑗∈𝐾 (𝑎 𝑖 − 𝑎 𝑗 )
Î
𝑥𝑖 ← 𝑧 · Î . (1.7)
𝑗∈𝐿 (𝑎 𝑖 − 𝑎 𝑗 )
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A few technical comments regarding the statement are in order. The first part of condition 1
in Theorem 1.8 generalizes the known property that SV𝑙 hits every multilinear polynomial that
contains a monomial of degree 𝑙 or less [48]. As for the second part, see Proposition 5.1 for
more discussion. The two parts together imply that all multilinear polynomials on 𝑛 ≤ 𝑘 + 𝑙 + 1
variables are hit by RFE𝑙𝑘 .
In condition 2, 𝜕𝐿 𝑝| 𝐾←0 denotes the polynomial obtained by taking the partial derivative
of 𝑝 with respect to every variable in 𝐿 and setting all the variables in 𝐾 to zero. Because of
the multilinearity, the order of the operations does not matter, and the resulting polynomial
depends only on variables in [𝑛] \ (𝐾 ∪ 𝐿). The substitution (1.7) can be viewed as 𝑥 𝑖 ← 𝑓 (𝑎 𝑖 ),
where
𝑗∈𝐾 (𝛼 − 𝑎 𝑗 )
Î
𝑓 (𝛼) = 𝑧 · 𝑓𝐾,𝐿 (𝛼) 𝑧 · Î
𝑗∈𝐿 (𝛼 − 𝑎 𝑗 )
is a valid seed of RFE𝑙𝑘 for polynomials in the variables 𝑥 𝑖 , 𝑖 ∈ [𝑛] \ (𝐾 ∪ 𝐿). Upon substitution,
𝜕𝐿 𝑝| 𝐾←0 becomes a univariate polynomial 𝑞 of degree at most 𝑛 − 𝑘 − 𝑙 in the fresh variable 𝑧.
In the case where 𝑝 is homogeneous, 𝑞 has at most one term, and 𝑞 is nonzero if and only 𝑞 is
nonzero at 𝑧 = 1. In general, for any fixed set 𝑍 of 𝑛 − 𝑘 − 𝑙 + 1 elements of 𝔽 , 𝑞 is nonzero if and
only if 𝑞 is nonzero at some 𝑧 ∈ 𝑍.
Theorem 1.8 can be understood as stating that a multilinear polynomial 𝑝 is hit by RFE𝑙𝑘
if and only if 𝑝 has a monomial supported on few or all-but-few variables, or else there is
a set of 𝑘 zero substitutions, 𝐾, and a set of 𝑙 partial derivatives, 𝐿, whose application to 𝑝
leaves a polynomial that is nonzero after substituting 𝑥 𝑖 ← 𝑧 · 𝑓𝐾,𝐿 (𝑎 𝑖 ). By judiciously choosing
variables for the zero substitutions and partial derivatives, prior papers managed to simplify
polynomials 𝑝 of certain types and reduce PIT for 𝑝 to PIT for simpler instances, resulting in
efficient recursive algorithms. In Section 5, we develop a general framework for such algorithms
and prove correctness directly from Theorem 1.8. Moreover, because Theorem 1.8 is a precise
characterization, any argument that SV or RFE hits a class of multilinear polynomials can be
converted into one within our framework, i. e., into an argument based on zero substitutions
and partial derivatives. Thus, Theorem 1.8 shows that these tools harness the complete power
of SV and RFE for multilinear polynomials.
Applications. We illustrate the utility of our characterizations of the vanishing ideal of RFE in
the two directions mentioned before.
Derandomization. To start, we demonstrate how Theorem 1.8 yields an alternate proof
of the result from [41] that SV1 —equivalently, RFE01 —hits every nonzero read-once formula
𝐹. Whereas the original proof hinges on a clever ad-hoc argument, our proof (described in
Section 5) is entirely systematic and amounts to a couple straightforward observations in order
to apply Theorem 1.8.
As a proof of concept of the additional power of our characterization for derandomization, we
make progress in a well-studied model for algebraic computation, namely read-once oblivious
algebraic branching programs (ROABPs). An ROABP consists of a layered digraph, the width
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of which constitutes an important complexity parameter. We refer to Section 8.1 for more
background.
Theorem 1.9 (ROABP hitting property). For any integer 𝑙 ≥ 1, SV𝑙 hits the class of polynomials
computed by read-once oblivious algebraic branching programs of width less than (𝑙/3) + 1 that contain a
monomial of degree at most 𝑙 + 1.
To the best of our knowledge, Theorem 1.9 is incomparable to the known results for ROABPs
[45, 29, 30, 17, 15, 3, 6, 26, 25, 24, 46, 10]. Without the restriction that the polynomial has a
monomial of degree at most 𝑙 + 1, Theorem 1.9 would imply a fully blackbox polynomial-time
identity test for the class of constant-width ROABPs. No such test has been proven to exist at
this time; prior work requires either quasipolynomial time or else opening the blackbox, such as
by knowing the order in which the variables are read.
With the restriction, hitting the class in Theorem 1.9 with SV𝑙 represents fairly specialized
progress. This is because SV𝑙+1 is well-known to hit every polynomial containing a monomial
of support 𝑙 + 1 or less, and thus hits the class in Theorem 1.9, irrespective of the restriction
on ROABP width. That said, the method of proof of Theorem 1.9 diverges significantly from
prior uses of the SV generator and therefore may be of independent interest. We elaborate
on the method more when we discuss the techniques of this paper, but for now, we point out
that most prior uses of the SV generator rely on combinatorial arguments, i. e., arguments that
depend only on which monomials are present in the polynomials to hit. Theorem 1.9 necessarily
goes beyond this because there is a polynomial in Van[SV𝑙 ] of degree 𝑙 + 1 that has the same
monomials as a polynomial computed by an ROABP of width 2, which by Theorem 1.9 is not in
Van[SV𝑙 ] for 𝑙 ≥ 4. Namely, any instance of ESMVC𝑙−1 𝑙 contains exactly all the monomials of the
form 𝑥 𝑖1 · 𝑥 𝑖2 · · · · · 𝑥 𝑖 𝑙+1 with (𝑖1 , . . . , 𝑖 𝑙+1 ) ∈ 𝑋1 × · · · × 𝑋𝑙+1 for some disjoint sets 𝑋 𝑗 ; the same
goes for 𝑗 𝑖 𝑗 ∈𝑋 𝑗 𝑥 𝑖 𝑗 , which is computed by an ROABP of width 2.
Î Í
Lower bounds. Our result for ROABPs also illustrates this direction. Our derandomization
result for the class in Theorem 1.9 is equivalent to the following lower bound.
Theorem 1.10 (ROABP lower bound). For any integer 𝑙 ≥ 1, any nonzero polynomial in Van[SV𝑙 ]
that contains a monomial of degree at most 𝑙 + 1, requires ROABP width at least (𝑙/3) + 1.
Such a lower bound is interesting because there are appealing polynomials meeting the
conditions, in particular the generators EVC𝑙−1
𝑙 as well as ESMVC𝑙−1
𝑙 . Other hardness of
representation results follow in a similar manner from prior hitting properties of SV in the
literature. The following lower bounds apply to computing both EVC𝑙𝑙−1 and ESMVC𝑙𝑙−1 :
• Any syntactically multilinear formula must have at least Ω(log(𝑙)/log log(𝑙)) reads of some
variable [7, Theorem 6.3].
• Any sum of read-once formulas must have at least Ω(𝑙) summands [41, Corollary 5.2].
• There exists an order of the variables such that any ROABP with that order must have
width at least 2Ω(𝑙) [15, Corollary 4.3].
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• Any Σm ΣΠ𝑂(1) formula must have top fan-in at least 2Ω(𝑙) [14]; see also [19, Lemma 5.12].
Ó
• Lower bounds over characteristic zero for circuits with locally-low algebraic rank [37,
Lemma 5.2].
Techniques. Many of our results ultimately require showing that, under suitable condi-
tions, RFE hits a polynomial 𝑝. A recurring analysis fulfills this role in the proofs of Theo-
rems 1.3, 1.8, and 1.9. We take intuition from the analytic setting (e. g., 𝔽 = ℝ) and study the
behavior of 𝑝(RFE) as a function of the seed’s zeroes and poles. When they are near the abscissas
of chosen variables of 𝑝, the behavior is dominated by the contributions of the monomials of 𝑝
for which the variables with abscissas near zeros have minimal degree and the variables with
abscissas near poles have maximal degree. This allows us to analyze a first approximation to
𝑝(RFE) by “zooming in” on the contributions of the monomials in which the chosen variables
have extremal degrees. If the first approximation is nonzero, then we can conclude that RFE
hits 𝑝. We capture the technique in our Zoom Lemma (Lemma 4.3). Formal Laurent series can
express the analytic intuition purely algebraically. We provide a proof from first principles that
does not require any background in Laurent series and works over all fields.
Theorem 1.3 states the equality 𝐼 = Van[RFE𝑙𝑘 ] of two ideals, where 𝐼 denotes the ideal
generated by all instantiations of EVC𝑙𝑘 , and Van[RFE𝑙𝑘 ] the vanishing ideal of RFE𝑙𝑘 .
• The inclusion ⊆ follows from linearizing the defining equations of RFE𝑙𝑘 (Lemma 3.1). The
technique mirrors the use of resultants to compute implicit equations for rational plane
curves. This is where the switch from SV to RFE helps.
• To establish the inclusion ⊇ we first show that the equivalence class of any polynomial 𝑝
modulo 𝐼 contains a representative 𝑟 whose monomials exhibit the combinatorial structure
of a core (Lemma 3.7). If 𝑝 ∉ 𝐼, 𝑟 is nonzero. The core structure of 𝑟 then allows us to apply
the zooming-in mechanism such that the resulting first approximation to 𝑟 is nonzero, in
which case the Zoom Lemma tells us that RFE𝑙𝑘 hits 𝑟 (Lemma 3.8). By the inclusion ⊆, we
conclude that RFE𝑙𝑘 hits 𝑝.
The proof of Theorem 1.8 also relies on the Zoom Lemma. Membership to the ideal is
equivalent to the vanishing of all coefficients of the expansion of 𝑝(RFE). The application of
the Zoom Lemma can be viewed as determining a small number of coefficients sufficient to
guarantee that their vanishing implies all coefficients vanish. The restriction to multilinear
polynomials 𝑝 allows us to express the zoomed-in contributions of 𝑝 as the result of applying
partial derivatives and zero-substitutions.
Theorem 1.9 makes use of the characterization of the minimum width of a read-once
oblivious algebraic branching program computing a polynomial 𝑝 as the maximum rank of the
monomial coefficient matrices of 𝑝 for various variable partitions [42]. The result is effectively
about polynomials 𝑝 that are homogeneous of degree 𝑙 + 1, in which case the monomial
coefficient matrices have a block-diagonal structure with 𝑙 + 2 blocks. An application of the
Zoom Lemma in the contrapositive yields linear equations between elements of consecutive
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blocks under the assumption that SV𝑙 fails to hit 𝑝. When some block is zero, the equations
yield a Cauchy system on the rows or columns of its neighboring blocks. Based on the fact
that Cauchy systems have full rank and exploiting the specific structure, we deduce several
constraints on the row-space/column-space of the neighboring blocks. A careful analysis and
case analysis based on the number of zero blocks yields a rank lower bound of at least (𝑙/3) + 1
for a well-chosen partition of the variables.
We point out that, in the preceding application, the Zoom Lemma is instantiated several
times in parallel to form a large system of equations on the coefficients of 𝑝, and the whole
system is necessary for the analysis. This stands in contrast to most prior work using SV, which
can be cast as using knowledge of how 𝑝 is computed to guide a search for a single fruitful
instantiation of the Zoom Lemma.
Alternating algebra representation. The inspiration for several of our results stems from
expressing the polynomials EVC𝑙𝑘 using concepts from alternating algebra (also known as
exterior algebra or Grassmann algebra). In fact, the membership test for the ideal generated by
the instantiations of EVC𝑙𝑘 in Theorem 1.8 is based on the relationship 𝜕2 = 0 from alternating
algebra. Our original statement and proof of the theorem made use of that framework, but
we managed to eliminate the alternating algebra afterwards. Still, as we find the perspective
insightful and potentially helpful for future developments, we describe the connection briefly
here and in more detail in Section 9. We explain the intuition for the simple case where the
degree of the polynomial 𝑝 equals 𝑙 + 1. In that setting, belonging to the ideal generated by the
polynomials EVC𝑙𝑘 is equivalent to being in their linear span.
The alternating algebra Λ∗ (𝑈) of a vector space 𝑈 over a field 𝔽 consists of the closure of 𝑈
under an additional binary operation, referred to as “wedge” and denoted ∧, which is bilinear,
associative, and satisfies
𝑢∧𝑢 =0 (1.8)
for every 𝑢 ∈ 𝑈. This determines a well-defined algebra. When the characteristic of 𝔽 is not 2,
(1.8) can equivalently be understood as anti-commutativity:
𝑢1 ∧ 𝑢2 = −(𝑢2 ∧ 𝑢1 ) (1.9)
for every 𝑢1 , 𝑢2 ∈ 𝑈. For any characteristic and 𝑢1 , 𝑢2 , . . . , 𝑢𝑡 ∈ 𝑈,
𝑢1 ∧ 𝑢2 ∧ · · · ∧ 𝑢𝑡 (1.10)
is nonzero iff the 𝑢𝑖 ’s are linearly independent, and any permutation of the order of the vectors in
(1.10) yields the same element of Λ∗ (𝑈) up to a sign. The sign equals the sign of the permutation,
whence the name “alternating algebra.” If 𝑈 has a basis 𝑉 = {𝑣 1 , . . . , 𝑣 𝑛 } of size 𝑛, then a basis
for Λ∗ (𝑈) can be formed by all 2𝑛 expressions of the form (1.10), where the 𝑢𝑖 ’s range over all
subsets of 𝑉 and are taken in some fixed order. Considering the elements of 𝑉 as vertices, the
basis elements of Λ∗ (𝑈) can be thought of as the oriented simplices of all dimensions that can
be built from 𝑉.
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Anti-commutativity arises naturally in the context of network flow, where 𝑉 denotes the
vertices of the underlying graph, and a wedge 𝑣1 ∧ 𝑣 2 of level 𝑡 = 2 represents one unit of flow
from 𝑣 1 to 𝑣2 . Equation (1.9) reflects the fact that one unit of flow from 𝑣 1 to 𝑣 2 cancels with one
unit of flow from 𝑣 2 to 𝑣 1 . The adjacent levels 𝑡 = 1 and 𝑡 = 3 also have natural interpretations
in the flow setting: 𝑣 1 (the element of Λ∗ (𝑈) of the form (1.10) with 𝑡 = 1) represents one unit of
surplus flow at 𝑣 1 (the vertex of the graph), and 𝑣 1 ∧ 𝑣 2 ∧ 𝑣 3 abstracts a circulation of one unit
along the directed cycle 𝑣 1 → 𝑣2 → 𝑣 3 → 𝑣 1 .
The different levels are related by so-called boundary maps, which are linear transformations
that map a simplex to a linear combination of its subsimplices of one dimension less. The maps
are parametrized by a linear weight function 𝑤 : 𝑈 → 𝔽 , and defined on the vertices by
𝑡
Õ
𝜕𝑤 : 𝑣 1 ∧ 𝑣 2 ∧ · · · ∧ 𝑣 𝑡 ↦→ (−1)𝑖+1 𝑤(𝑣 𝑖 ) 𝑣 1 ∧ · · · ∧ 𝑣 𝑖−1 ∧ 𝑣 𝑖+1 ∧ · · · ∧ 𝑣 𝑡 , (1.11)
𝑖=1
an expression resembling the minor expansion of a determinant along a column [𝑤(𝑣 𝑖 )]𝑡𝑖=1 . In
the flow setting, using 𝑤 ≡ 1, applying 𝜕1 to 𝑣1 ∧ 𝑣 2 yields 𝑣 2 − 𝑣 1 , the superposition of demand
at 𝑣 1 and surplus at 𝑣 2 corresponding to one unit of flow from 𝑣 1 to 𝑣 2 . Likewise, 𝜕1 sends the
abstract elementary circulation 𝑣1 ∧ 𝑣 2 ∧ 𝑣 3 to the superposition of the three edge flows that
make up the directed 3-cycle 𝑣1 → 𝑣 2 → 𝑣 3 → 𝑣 1 . A linear combination 𝑝 of terms (1.10) with
𝑡 = 2 represents a valid circulation iff it satisfies conservation of flow at every vertex, which
can be expressed as 𝜕1 (𝑝) = 0, i. e., 𝑝 is in the kernel of 𝜕1 . An equivalent criterion is for 𝑝 to be
the superposition of circulations around directed 3-cycles, which can be expressed as 𝑝 being
in the image of 𝜕1 . The relationship im(𝜕𝑤 ) = ker(𝜕𝑤 ) between the image and the kernel of a
boundary map holds for any nonzero 𝑤, and generalizes to composed boundary maps: For any
linearly independent 𝑤1 , . . . , 𝑤 𝑘+1 , it holds that
𝑘+1
Ù
im 𝜕𝑤 𝑘+1 ◦ 𝜕𝑤 𝑘 ◦ · · · ◦ 𝜕𝑤1 = ker (𝜕𝑤 𝑟 ) .
(1.12)
𝑟=1
When 𝑤1 , . . . , 𝑤 𝑘+1 are linearly dependent, 𝜕𝑤 𝑘+1 ◦ · · · ◦ 𝜕𝑤1 is the zero map.
In the context of RFE, the set 𝑉 consists of a distinct vertex 𝑣 𝑖 for each variable 𝑥 𝑖 , and
simplices correspond to multilinear monomials. The anti-commutativity of ∧ coincides with the
fact that swapping two arguments to EVC𝑙𝑘 means swapping two rows in (1.5), which changes
the sign of the determinant. Using boundary maps, EVC𝑙𝑘 [𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+2 ] can be viewed as
𝜕𝜔 (𝑣 𝑖1 ∧ 𝑣 𝑖2 ∧ · · · ∧ 𝑣 𝑖 𝑘+𝑙+2 ), where 𝜕𝜔 𝜕𝑤 𝑘+1 ◦ 𝜕𝑤 𝑘 ◦ · · · ◦ 𝜕𝑤1 and 𝑤 𝑟 (𝑣 𝑖 ) (𝑎 𝑖 )𝑟−1 . By (1.12), this
means that EVC𝑙𝑘 is in the kernel of 𝜕𝑤 𝑟 for each 𝑟 ∈ [𝑘 + 1], or equivalently, in the kernel of 𝜕𝑤 for
each 𝑤 : 𝑈 → 𝔽 of the form 𝑤(𝑣 𝑖 ) = 𝑞(𝑎 𝑖 ) where 𝑞 is a polynomial of degree at most 𝑘. In fact,
(1.12) implies that the linear span of the generators EVC𝑙𝑘 consists exactly of the polynomials of
degree 𝑙 + 1 in this kernel. The linear span coincides with the polynomials of degree 𝑙 + 1 in
the ideal generated by the polynomials EVC𝑙𝑘 . For multilinear polynomials, being in the kernel
can be expressed in terms of zero substitutions and partial derivatives as in Theorem 1.8. This
yields an alternate route for deriving our membership test for multilinear polynomials of degree
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𝑑 = 𝑙 + 1 in the ideal generated by the instantiations of EVC𝑙𝑘 , which by Theorem 1.3 agrees with
Van[RFE𝑙𝑘 ]. In the basic case where 𝑘 = 0 and 𝑙 = 1, only the weight function 𝑤 ≡ 1 needs to be
considered and the kernel requirement coincides with flow conservation. We refer to Section 9
for the general multilinear case of arbitrary degree.
Related recent work and further research. We propose to systematically investigate the power
of generators by characterizing their vanishing ideals. As we demonstrated for SV and RFE,
such characterizations can exhibit both strengths and weaknesses of the generator.
Specific other generators of interest include Klivans–Spielman [34] and generators based
on the matrix rank condenser by Gabizon and Raz [21, 33, 16]. A related direction is figuring
out how vanishing ideals are affected when manipulating generators. Examples include the
RFE generator with pseudorandom abscissas, or work that relates the vanishing ideal of a
combination of generators to the vanishing ideals of the constituent generators. In particular, a
combination of SV with Klivans–Spielman appears in the literature [32, 15, 19], where the latter
is used to effectively hit sparse polynomials, which our results show that SV does not.
The generator SV𝑙 is the canonical example of an 𝑙-wise independent generator in the
algebraic setting. Understanding the power of 𝑙-wise independent generators more broadly,
e. g., as formalized in [18, 39], could lead to useful insights for derandomizing PIT. This work
demonstrates explicit polynomials like EVC𝑙−1 𝑙 and ESMVC𝑙−1𝑙 that are not automatically hit by
𝑙
𝑙-wise independence as they are not hit by SV . Is there a deeper underlying reason related to
𝑙-wise independence?
A generator hits all polynomials from a resource-bounded class iff no nonzero polynomial in
the vanishing ideal can be computed within those resources. Chatterjee and Tengse [11] recently
showed the following generic limitation: The vanishing ideal of any generator computable by
algebraic circuits of polynomial size in the number of variables contains a nonzero polynomial
computable in VPSPACE. From this perspective, our results exhibit a weakness of SV and RFE
in that their vanishing ideals contain nonzero polynomials from the presumably much smaller
class VBP. In fact, EVC𝑙𝑘 is a polynomial depending on only 𝑘 + 𝑙 + 2 variables and is computable
by a branching program of size polynomial in the number of variables. Thus, in order to hit all
branching programs of size 𝑠, SV and RFE require a seed length 𝑘 + 𝑙 + 2 = 𝑠 Ω(1) .
A related question is whether the generators we have identified have minimal (or approxi-
mately minimal) complexity in the vanishing ideal. Andrews and Forbes [8] recently established
such a result for a generator that substitutes an 𝑛 × 𝑚 matrix of variables with the product
of 𝑛 × 𝑙 and 𝑙 × 𝑚 matrices of variables for small 𝑙. The vanishing ideal of their generator is
straightforwardly generated by (𝑙 + 1) × (𝑙 + 1) minors. For this vanishing ideal the authors
manage to show that every nonzero element is at least as hard as computing Θ(𝑙 1/3 ) × Θ(𝑙 1/3 )
determinants (under simple reductions and in the sense of border complexity).
Lastly, we list some avenues for improving specific aspects of our results. Theorem 1.8
represents an elementary deterministic membership test in the vanishing ideal of RFE𝑙𝑘 for
multilinear polynomials. Can the elementary test can be extended to all polynomials? From
the alternating algebra perspective, the test relies an the convenient one-to-one correspondence
between multilinear polynomials and elements of the alternating algebra. For general poly-
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
nomials, this correspondence is no longer one-to-one, and the resulting membership test is
nondeterministic.
Another target is eliminating degree restrictions for our characterizations of specialized
classes of polynomials, in particular in Theorem 1.9 for ROABPs. Removing the degree restriction
for ROABPs would result in a full blackbox derandomization of constant-width ROABPs. An
alternative possibility is that, through better analysis of the vanishing ideal, it turns out that
RFE has limitations in derandomizing constant-width ROABPs.
Organization. We start in Section 2 with formal aspects of the RFE generator that have
been omitted from the informal discussion thus far. We construct the generating set for the
vanishing ideal (Theorem 1.3) in Section 3, followed by the Zoom Lemma in Section 4. The ideal
membership test (Theorem 1.8) is developed in Section 5. We present the results on sparseness
in Section 6, and the ones on set-multilinearity in Section 7. Background on ROABPs and our
result on derandomizing PIT for ROABPs (Theorem 1.9) are covered in Section 8. We end our
paper in Section 9 with a further discussion of the alternating algebra representation and an
alternate derivation of the membership test for multilinear polynomials in the ideal generated
by the instances of EVC𝑙𝑘 .
2 RFE Generator
We defined the RFE generator in the overview but omitted some of the formal details. In this
section, we discuss different parametrizations of RFE as well as how to obtain deterministic
blackbox PIT algorithms from a generator and how large the underlying field 𝔽 must be. We
also state and establish the precise relationship between RFE𝑙−1 𝑙 and SV𝑙 .
In Definition 1.2, we defined RFE as a set of substitutions formed by varying the seed 𝑓
over certain rational functions with coefficients in 𝔽 . Meanwhile, our analyses proceed by
parametrizing 𝑓 by scalars, abstracting the scalar parameters as fresh formal variables, and
calculating in the field of rational functions in those variables. The approaches are equivalent
over large enough fields, and the flexibility to choose is a source of convenience. Here are some
natural parametrizations of 𝑓 :
Coefficients. Select scalars 𝑔0 , . . . , 𝑔 𝑘 , ℎ0 , . . . , ℎ 𝑙 ∈ 𝔽 and set
𝑔 𝑘 𝛼 𝑘 + 𝑔 𝑘−1 𝛼 𝑘−1 + · · · + 𝑔1 𝛼 + 𝑔0
𝑓 (𝛼) = ,
ℎ 𝑙 𝛼 𝑙 + ℎ 𝑙−1 𝛼 𝑙−1 + · · · + ℎ 1 𝛼 + ℎ0
ignoring choices of ℎ 0 , . . . , ℎ 𝑙 for which the denominator vanishes at some abscissa.
Evaluations. Fix two collections, 𝐵 = {𝑏 1 , . . . , 𝑏 𝑘+1 } and 𝐶 = {𝑐1 , . . . , 𝑐 𝑙+1 }, each of distinct
scalars from 𝔽 . Then select scalars 𝑔1 , . . . , 𝑔 𝑘+1 and ℎ 1 , . . . , ℎ 𝑙+1 and set
𝑔(𝛼)
𝑓 (𝛼) =
ℎ(𝛼)
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
where 𝑔 is the unique degree-𝑘 polynomial with 𝑔(𝑏1 ) = 𝑔1 , 𝑔(𝑏2 ) = 𝑔2 , . . . , 𝑔(𝑏 𝑘+1 ) = 𝑔 𝑘+1 ,
and ℎ is defined similarly with respect to 𝐶. Choices of ℎ1 , . . . , ℎ 𝑙+1 that lead ℎ to vanish
at some abscissa are ignored.
Note that an explicit formula for 𝑔 and ℎ in terms of the parameters can be obtained using
the Lagrange interpolants with respect to 𝐵 and 𝐶.
Roots. Select scalars 𝑧, 𝑠1 , . . . , 𝑠 𝑘0 , 𝑡1 , . . . , 𝑡 𝑙0 ∈ 𝔽 for some 𝑘 0 ≤ 𝑘 and 𝑙 0 ≤ 𝑙 and set
(𝛼 − 𝑠 1 ) · · · · · (𝛼 − 𝑠 𝑘0 )
𝑓 (𝛼) = 𝑧 · ,
(𝛼 − 𝑡1 ) · · · · · (𝛼 − 𝑡 𝑙0 )
where {𝑡1 , . . . , 𝑡 𝑙0 } is disjoint from the set of abscissas.
In fact, it is no loss of power to restrict to 𝑘 0 = 𝑘 and 𝑙 0 = 𝑙.
Hybrids are of course possible, too. For example, Proposition 2.2 below uses the evaluations
parametrization for the numerator and roots parametrization for the denominator.
The following lemma justifies that, for any polynomial 𝑝, as long as 𝔽 is large enough, 𝑝(RFE)
vanishes with respect to a particular parametrization of RFE if and only if it vanishes with
respect to RFE as defined in Definition 1.2. The lemma is an immediate consequence of the
well-known analogous result for polynomials, sometimes referred to as the Polynomial Identity
Lemma [44, 13, 50, 47, 9].
Lemma 2.1. Let 𝔽 be field, and 𝑓 = 𝑔/ℎ ∈ 𝔽 (𝜏1 , . . . , 𝜏𝑙 ) be a rational function in 𝑙 variables with
deg(𝑔) ≤ 𝑑 and deg(ℎ) ≤ 𝑑. Let 𝑆 ⊆ 𝔽 be finite. Then the probability that 𝑓 vanishes or is undefined
when each 𝜏𝑖 is substituted by a uniformly random element of 𝑆 is at most 2𝑑/|𝑆|.
Proof. The rational function 𝑓 vanishes or is undefined if and only if the polynomial 𝑝 = 𝑔 ℎ
vanishes, which happens with probability at most deg(𝑝)/|𝑆| according to the Polynomial
Identity Lemma.
In particular, if 𝔽 is infinite, then, for all polynomials 𝑝, all the above parametrizations and
Definition 1.2 are equivalent for the purposes of hitting 𝑝; when 𝑝 is fixed, the equivalence holds
provided | 𝔽 | ≥ poly(𝑛, deg(𝑝)). Quantitative bounds on the number of substitutions to perform
when testing whether RFE hits 𝑝 in the blackbox algorithm likewise follow from Lemma 2.1. As
is customary in the context of blackbox derandomization of PIT, if 𝔽 is not large enough, then
one works instead over a sufficiently large extension of 𝔽 .
We now formally state and argue the close relationship between RFE𝑙−1
𝑙 and SV𝑙 that we
sketched in Section 1.
Proposition 2.2. Let 𝑙 and 𝑛 be positive integers. There is an invertible diagonal transformation 𝐴 :
𝔽 𝑛 → 𝔽 𝑛 such that, for any polynomial 𝑝 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ], 𝑝(SV𝑙 ) = 0 if and only if (𝑝 ◦𝐴)(RFE𝑙−1
𝑙 ) = 0.
In particular, the vanishing ideals of RFE𝑙−1
𝑙 and of SV𝑙 are the same up to the rescaling of
Proposition 2.2.
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Proof of Proposition 2.2. Let 𝔽b be the field of rational functions in indeterminates 𝜐1 , . . . , 𝜐 𝑙 , 𝜁1 ,
. . . , 𝜁 𝑙 over 𝔽 . A polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] has 𝑝(SV𝑙 ) = 0 if and only if 𝑝 vanishes at the
point
𝑙 Ö 𝜐𝑡 − 𝑎 𝑗
©Õ
𝜁𝑡 : 𝑖 ∈ [𝑛]® ∈ 𝔽b𝑛 .
ª
(2.1)
𝑎𝑖 − 𝑎 𝑗
« 𝑡=1 𝑗∈[𝑛]\{𝑖} ¬
Set 𝐴 : 𝔽 𝑛 → 𝔽 𝑛 to be the diagonal linear transformation that divides the coordinate for 𝑥 𝑖 by
𝑗∈[𝑛]\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 ). 𝐴 is invertible. Applying 𝐴
Î −1 to (2.1) yields the point
𝑙 𝑙
©Õ Ö ª ©Õ © Ö ª 1
𝜁𝑡 (𝜐𝑡 − 𝑎 𝑗 ) : 𝑖 ∈ [𝑛]® = 𝜁𝑡 (𝜐𝑡 − 𝑎 𝑗 )® : 𝑖 ∈ [𝑛]® .
ª
(2.2)
𝜐𝑡 − 𝑎 𝑖
« 𝑡=1 𝑗∈[𝑛]\{𝑖} ¬ « 𝑡=1 « 𝑗∈[𝑛] ¬ ¬
𝑝 vanishes at (2.1) if and only if 𝑝 ◦ 𝐴 vanishes at (2.2). Now let 𝔽b0 be the field of rational
functions in indeterminates 𝜏1 , . . . , 𝜏𝑙 , 𝜎1 , . . . , 𝜎𝑙 over 𝔽 . After the change of variables
1 −𝜎𝑡
𝜁𝑡 ← Î ·Î and 𝜐𝑡 ← 𝜏𝑡
𝑗∈[𝑛] (𝜏𝑡 − 𝑎 𝑗 ) 𝑠≠𝑡 (𝜏𝑡 − 𝜏𝑠 )
(2.2) becomes
𝑙
Í𝑙 𝑎 𝑖 −𝜏𝑠
𝑡=1 𝜎𝑡
! Î !
Õ 𝜎𝑡 1 𝑠≠𝑡 𝜏𝑡 −𝜏𝑠
: 𝑖 ∈ [𝑛] = : 𝑖 ∈ [𝑛] ∈ 𝔽b0𝑛 . (2.3)
𝑠≠𝑡 𝜏𝑡 − 𝜏𝑠 𝑎 𝑖 − 𝜏𝑡
Î𝑙
𝑡=1 𝑎 𝑖 − 𝜏𝑡
Î
𝑡=1
Since the change of variables is invertible, 𝑝 ◦ 𝐴 vanishes at (2.2) if and only if it vanishes at (2.3).
Now, viewing 𝜎1 , . . . , 𝜎𝑙 , 𝜏1 , . . . , 𝜏𝑙 as seed variables, observe that the right-hand side of (2.3)
is RFE𝑙−1
𝑙 (𝑔/ℎ) where 𝑔 is parametrized by evaluations (𝑔(𝜏𝑡 ) = 𝜎𝑡 ) and ℎ is parametrized by
roots (𝜏1 , . . . , 𝜏𝑙 ). It follows that 𝑝 ◦ 𝐴 vanishes at (2.3) if and only if (𝑝 ◦ 𝐴)(RFE𝑙𝑙−1 ) = 0.
3 Generating Set
In this section, we establish Theorem 1.3, our characterization of the vanishing ideal of RFE
in terms of an explicit generating set. For every 𝑘, 𝑙 ∈ ℕ , we develop a template, EVC𝑙𝑘 , for
constructing polynomials that belong to the vanishing ideal of RFE𝑙𝑘 such that all instantiations
collectively generate the vanishing ideal.
The template can be derived in the following fashion. Fix any seed 𝑓 of RFE𝑙𝑘 , and write it as
Í𝑘
𝑓 = 𝑔/ℎ where 𝑔(𝛼) = 𝑑=0 𝑔 𝑑 𝛼 𝑑 and ℎ(𝛼) = 𝑙𝑑=0 ℎ 𝑑 𝛼 𝑑 are respectively polynomials of degree
Í
𝑘 and 𝑙. For any 𝑖 ∈ [𝑛], the polynomial 𝑔(𝑎 𝑖 )/ℎ(𝑎 𝑖 ) − 𝑥 𝑖 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] vanishes by definition at
RFE𝑙𝑘 ( 𝑓 ). While this polynomial varies with 𝑓 , it does so uniformly. Specifically, after rescaling
to 𝑔(𝑎 𝑖 ) − ℎ(𝑎 𝑖 )𝑥 𝑖 , the polynomial depends only linearly on the coefficients of 𝑔 and ℎ. We exploit
this uniformity to construct a polynomial that vanishes at RFE𝑙𝑘 ( 𝑓 ) but that now is independent
of 𝑓 . Since 𝑓 is arbitrary, the constructed polynomial belongs to the vanishing ideal of RFE𝑙𝑘 .
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The construction begins by expressing the vanishing of each 𝑔(𝑎 𝑖 ) − ℎ(𝑎 𝑖 )𝑥 𝑖 at RFE𝑙𝑘 ( 𝑓 ) as
the following system of equations. Abbreviating
|
𝑔® 𝑔 𝑘 𝑔 𝑘−1 . . . 𝑔1 𝑔0
|
ℎ® ℎ 𝑙 ℎ 𝑙−1 . . . ℎ1 ℎ0 ,
we write
𝑔®
𝑘
𝑎𝑖 𝑎 𝑖𝑘−1 ... 𝑎 𝑖𝑙 𝑥 𝑖 𝑎 𝑖𝑙−1 𝑥 𝑖 ... 𝑥 𝑖 𝑖∈[𝑛] ·
1 = 0. (3.1)
− ℎ®
Written this way, (3.1) has the form of a homogeneous system of linear equations. There is one
equation for each 𝑖 ∈ [𝑛] and one unknown for each of the 𝑘 + 𝑙 + 2 parameters of the seed 𝑓 .
The system’s coefficient matrix has no dependence on 𝑓 , but for any 𝑓 , substituting RFE𝑙𝑘 ( 𝑓 )
into 𝑥 1 , . . . , 𝑥 𝑛 yields a system that has a nontrivial solution, namely the vector in (3.1).
Consider, then, the determinant of the square subsystem of (3.1) formed by any 𝑘 + 𝑙 + 2
rows. It is a polynomial in 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ]. Because the coefficient matrix in (3.1) is independent
of 𝑓 , the determinant is independent of 𝑓 . Because the subsystem has a nonzero solution after
substituting RFE𝑙𝑘 ( 𝑓 ) for any 𝑓 , the determinant vanishes after substituting RFE𝑙𝑘 ( 𝑓 ) for any 𝑓 .
We conclude that the determinant belongs to the vanishing ideal of RFE𝑙𝑘 .
Recalling that the determinant for the subsystem consisting of rows 𝑖1 , . . . , 𝑖 𝑘+𝑙+2 is identically
EVC𝑙𝑘 [𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+2 ], we have established:
Lemma 3.1. For every 𝑘, 𝑙 ∈ ℕ and 𝑖1 , 𝑖2 , . . . , 𝑖 𝑘+𝑙+2 ∈ [𝑛], EVC𝑙𝑘 [𝑖1 , . . . , 𝑖 𝑘+𝑙+2 ] ∈ Van[RFE𝑙𝑘 ].
As we explain at the end of this section, the above derivation is where our use of RFE in
lieu of SV plays a critical role. Before moving on, we also point a few elementary properties
and an provide an explicit expression for the coefficients of EVC𝑙𝑘 as products of Vandermonde
determinants in the abscissas and a sign term. We introduce the following notation for the
underlying Vandermonde matrices.
Definition 3.2 (Vandermonde matrix). For 𝑇 = {𝑖1 , . . . , 𝑖 𝑡 } ⊆ [𝑛] with 𝑖1 < · · · < 𝑖 𝑡 , we abbreviate
the Vandermonde matrix built from 𝑎 𝑖 for 𝑖 ∈ 𝑇 in increasing order as
𝑎 𝑡−1 · · · 1
𝑖1
𝐴𝑇 ... .. .
.
(3.2)
𝑡−1
𝑎 ··· 1
𝑖𝑡
The sign term makes use of the following standard combinatorial quantity.
Definition 3.3 (cross inversions). For 𝐴, 𝐵 ⊆ [𝑛], XInv(𝐴, 𝐵) |{(𝑎, 𝑏) ∈ 𝐴 × 𝐵 | 𝑎 > 𝑏}| denotes
the number of cross inversions between 𝐴 and 𝐵.
Proposition 3.4. For every 𝑘, 𝑙 ∈ ℕ , EVC𝑙𝑘 is skew-symmetric in that, for any 𝑖1 , . . . , 𝑖 𝑘+𝑙+2 ∈ [𝑛] and
permutation 𝜋 of [𝑘 + 𝑙 + 2],
EVC𝑙𝑘 [𝑖1 , . . . , 𝑖 𝑘+𝑙+2 ] = sign(𝜋) · EVC𝑙𝑘 [𝑖𝜋(1) , . . . , 𝑖 𝜋(𝑘+𝑙+2) )].
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For any 𝑆 ⊆ [𝑛] with |𝑆| = 𝑘 + 𝑙 + 2, EVC𝑙𝑘 [𝑆] is a nonzero, multilinear, and homogeneous polynomial
of total degree 𝑙 + 1, and every multilinear monomial of degree 𝑙 + 1 in 𝑥 𝑖1 , . . . , 𝑥 𝑖 𝑘+𝑙+2 appears with a
nonzero coefficient. More specifically, for 𝑆 = {𝑖1 , . . . , 𝑖 𝑘+𝑙+2 } ⊆ [𝑛] with 𝑖1 < 𝑖2 < . . . , 𝑖 𝑘+𝑙+2 ,
Õ Ö
EVC𝑙𝑘 [𝑆] = 𝛾𝐾,𝐿 · 𝑥𝑖 , (3.3)
𝐾t𝐿=𝑆 𝑖∈𝐿
|𝐿|=𝑙+1
where
𝛾𝐾,𝐿 (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · det(𝐴𝐿 ). (3.4)
Proof. All assertions follow from elementary properties of determinants, that Vandermonde
determinants are nonzero unless they have duplicate rows, and the following computation. The
coefficient 𝛾𝐾,𝐿 can be obtained by plugging in 0 for 𝑥 𝑖 with 𝑖 ∈ 𝐾, and 1 for 𝑥 𝑖 with 𝑖 ∈ 𝐿. For
𝐾 ∗ consisting of the first 𝑘 + 1 elements of 𝑆 and 𝐿∗ of the last 𝑙 + 1, this yields the determinant
𝑎 𝑖𝑘1 ··· 1 0 ··· 0
.. .. .. .. .. ..
. . . . . .
𝑎 𝑖𝑘 ··· 1 0 ··· 0
𝑘+1
, (3.5)
∗ ··· ∗ 𝑎 𝑖𝑙 ··· 1
𝑘+2
.. .. .. .. .. ..
. . . . . .
∗ ··· ∗ 𝑎 𝑖𝑙 ··· 1
𝑘+𝑙+2
which equals the product of the Vandermonde matrices det(𝐴𝐾 ∗ ) and det(𝐴𝐿∗ ), and confirms the
expression for 𝛾𝐾 ∗ ,𝐿∗ as XInv(𝐾 ∗ , 𝐿∗ ) = 0. For general 𝐾 and 𝐿, we obtain a determinant with the
same shape as (3.5) after rearranging the rows such that the ones involving 𝑎 𝑖 for 𝑖 ∈ 𝐾 appear
first and in order, and the ones involving 𝑎 𝑖 for 𝑖 ∈ 𝐿 appear last and in order. By skew-symmetry,
the rearrangement induces an additional factor of sign(𝜋) = (−1)Inv(𝜋) , where 𝜋 denotes the
underlying permutation of [𝑘 + 𝑙 + 2] and Inv(𝜋) denotes the number of inversions of 𝜋, which
equals XInv(𝐾, 𝐿).
Lemma 3.1 shows that the polynomials EVC𝑙𝑘 [𝑖1 , . . . , 𝑖 𝑘+𝑙+2 ] belong to the vanishing ideal
of RFE𝑙𝑘 . In fact, various subsets of them generate the vanishing ideal. To prove that a certain
subset does so, we establish the following two steps:
1. Modulo the ideal 𝐼 generated by the subset, every polynomial 𝑝 is equal to a polynomial 𝑟
with a particular combinatorial structure (Lemma 3.7).
2. Every nonzero polynomial 𝑟 with that structure is hit by RFE𝑙𝑘 (Lemma 3.8).
Together, these show that every polynomial in the vanishing ideal of RFE𝑙𝑘 is equal to the zero
polynomial modulo the ideal 𝐼. We conclude that the ideals coincide, i. e., the vanishing ideal of
RFE𝑙𝑘 is generated by the subset of instantiations of EVC𝑙𝑘 that define 𝐼.
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For the subset of instantiations EVC𝑙𝑘 [𝐶 t 𝐿] where 𝐶 ⊆ [𝑛] is any fixed subset of size 𝑘 + 1
and 𝐿 ⊆ [𝑛] ranges over all subsets of size 𝑙 + 1 disjoint from 𝐶, the combinatorial structure
bridging the two steps is that the polynomial is cored.
Definition 3.5 (monomial support and cored polynomial). The support of a monomial 𝑚 ∈
𝔽 [𝑥1 , . . . , 𝑥 𝑛 ], denoted supp(𝑚), is the set of indices 𝑖 ∈ [𝑛] such that 𝑚 depends on 𝑥 𝑖 . For
𝑐, 𝑡 ∈ ℕ , a polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] is said to be (𝑐, 𝑡)-cored if there exists 𝐶 ⊆ [𝑛], called
the core, such that |𝐶| ≤ 𝑐 and for every monomial 𝑚 of 𝑝, |supp(𝑚) \ 𝐶| ≤ 𝑡. For any subset
𝐶 ⊆ [𝑛] and monomial 𝑚 = 𝑖∈[𝑛] 𝑥 𝑖𝑑𝑖 , we call 𝑖∈𝐶 𝑥 𝑖𝑑𝑖 the 𝐶-part of 𝑚, and 𝑖∈[𝑛]\𝐶 𝑥 𝑖𝑑𝑖 the
Î Î Î
non-𝐶-part of 𝑚.
The crux for the first step is the following property, which allows us to gradually get closer
to a (𝑘 + 1, 𝑙)-cored polynomial.
Proposition 3.6. Let 𝑘, 𝑙, 𝑛 ∈ ℕ , let 𝐶 be a (𝑘 + 1)-subset of [𝑛], and let 𝐼 denote the ideal generated by
the polynomials EVC𝑙𝑘 [𝐶 t 𝐿] where 𝐿 ranges over all (𝑙 + 1)-subsets of [𝑛] \ 𝐶. Consider a monomial
𝑚 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] such that |supp(𝑚) \ 𝐶| > 𝑙. Modulo 𝐼, 𝑚 is equal to a linear combination of
monomials whose non-𝐶-parts have lower degree than the non-𝐶-part of 𝑚.
Proof. Let 𝐿 be a subset of supp(𝑚) \ 𝐶 of size 𝑙 + 1. Let 𝑚 0 be the monomial such that 𝑚 = 𝑚 0 · 𝑥 𝐿 ,
where 𝑥 𝐿 𝑖∈𝐿 𝑥 𝑖 . By Proposition 3.4, 𝑥 𝐿 is a monomial of EVC𝑙𝑘 [𝐶 t 𝐿], and every other
Î
monomial of EVC𝑙𝑘 [𝐶 t 𝐿] has non-𝐶-part of degree at most 𝑙. It follows that 𝑚 0 · EVC𝑙𝑘 [𝐶 t 𝐿]
can be written as 𝑐 · 𝑚 + 𝑟, where 𝑐 in a nonzero element in 𝔽 and every monomial in 𝑟 has
non-𝐶-part of lower degree than 𝑚 does. Since ideals are closed under multiplication by any
other polynomial, 𝑚 0 · EVC𝑙𝑘 [𝐶 t 𝐿] ∈ 𝐼. Thus, we have 0 ≡ 𝑐 · 𝑚 + 𝑟 mod 𝐼, which can be
rewritten as 𝑚 ≡ −𝑐 −1 · 𝑟 mod 𝐼.
Proposition 3.6 leads to the following formalization of the first step of our approach.
Lemma 3.7. Let 𝑘, 𝑙, 𝑛 ∈ ℕ , let 𝐶 be a (𝑘 + 1)-subset of [𝑛], and let 𝐼 be the ideal generated by the
polynomials EVC𝑙𝑘 [𝐶 t 𝐿] where 𝐿 ranges over all (𝑙 + 1)-subsets of [𝑛] \ 𝐶. Modulo 𝐼, every polynomial
is equal to a (𝑘 + 1, 𝑙)-cored polynomial with core 𝐶.
Proof. For any polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], Proposition 3.6 allows us to systematically eliminate
any monomial 𝑚 in 𝑝 that violates the (𝑘 + 1, 𝑙)-cored condition, without changing 𝑝 modulo
𝐼. The process may introduce other monomials, but those monomials all have non-𝐶-parts of
degree lower than 𝑚 does. This means that the process cannot continue indefinitely. When it
ends, the remaining polynomial is (𝑘 + 1, 𝑙)-cored with core 𝐶 and is equivalent to 𝑝 modulo
𝐼.
The second step of our approach is formalized in Lemma 3.8.
Lemma 3.8. Let 𝑘, 𝑙, 𝑛 ∈ ℕ and let 𝑟 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] be nonzero and (𝑘 + 1, 𝑙)-cored. Then RFE𝑙𝑘 hits
𝑟.
We prove Lemma 3.8 from the Zoom Lemma in Section 4. Assuming it, we have all
ingredients for the proof of Theorem 1.3.
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
Proof of Theorem 1.3. The combination of Lemma 3.1, Lemma 3.7, and Lemma 3.8 shows that, for
every core 𝐶 ⊆ [𝑛] of size 𝑘 + 1, the vanishing ideal Van[RFE𝑙𝑘 ] is generated by the polynomials
EVC𝑙𝑘 [𝐶 t 𝐿] where 𝐿 ranges over all (𝑙 +1)-subsets of [𝑛]\ 𝐶. The generators are all homogeneous
of minimum degree 𝑙 + 1, and each generator has a monomial that occurs in none of the other
generators (namely the product of the variables in 𝐿). Therefore, the generating set has minimum
size since it forms a vector space basis of the degree-(𝑙 + 1) part of Van[RFE𝑙𝑘 ].
As an aside, we justify along the same lines the claim from Section 1 that all linear
dependencies among instances of EVC𝑙𝑘 are generated by the equations (1.6) when {𝑖1 , . . . , 𝑖 𝑘+𝑙+3 }
ranges over all subsets of [𝑛] containing a core 𝐶 of size 𝑘 + 1. A similar reduction strategy
modulo those equations allows us to rewrite
Õ
𝑐 𝑆 · EVC𝑙𝑘 [𝑆] = 0
𝑆⊆[𝑛]
|𝑆|=𝑘+𝑙+2
such that the range of the subsets 𝑆 is reduced to a (𝑘 + 1, 𝑙 + 1)-cored subclass with core 𝐶. By
linear independence, the only equation of that form is the trivial one with all 𝑐 𝑆 = 0.
Gröbner basis. We end this section with a short discussion on Gröbner bases. This part is
not essential for understanding the remainder of the paper; the reader may feel free to skip it.
Readers who want to know more may refer to [1, 12].
Gröbner bases are useful for solving several computational problems involving ideals,
including determining whether a given polynomial 𝑝 belongs to an ideal 𝐼 given by a finite set 𝐺
of generators. The setup presumes a total order ≥ on monomials with the following properties:
• For all monomials 𝑚, we have 𝑚 ≥ 1, where 1 denotes the empty monomial.
• For monomials 𝑚1 , 𝑚2 , 𝑚, we have that if 𝑚1 ≥ 𝑚2 , then 𝑚1 · 𝑚 ≥ 𝑚2 · 𝑚.
Assuming such a monomial ordering ≥, every nonzero polynomial has a unique monomial that
is maximal in ≥, which we call the leading monomial.
For a given a polynomial 𝑝, we can compute a 𝐺-reduced form of 𝑝 by repeatedly applying
the following reduction step, starting from 𝑓 = 𝑝: Find 𝑔 ∈ 𝐺 such that the leading monomial
of 𝑔 divides some monomial 𝑚 of 𝑓 , and then subtract a suitable multiple of 𝑔 from 𝑓 so as to
produce a new value of 𝑓 that does not contain 𝑚 as a monomial. If several such 𝑔 and 𝑚 exist,
pick any. The process continues until no suitable 𝑔 and 𝑚 can be found, which the properties of
the ordering ≥ guarantee to happen at some point. The final 𝑓 is called a 𝐺-reduced form of 𝑝,
which may or may not be unique.
A natural algorithm to determine membership of 𝑝 in 𝐼 is to compute a 𝐺-reduced form
𝑓 of 𝑝 and conclude that 𝑝 ∈ 𝐼 if and only if 𝑓 = 0. The algorithm generalizes computing
the remainder in univariate polynomial division, with some differences being that there are
multiple choices of 𝑔, and the monomial 𝑚 does not need to be the leading monomial of 𝑓 . A
positive conclusion, 𝑝 ∈ 𝐼, is always correct because reduction does not affect membership and 0
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
trivially belongs to the ideal. However, the algorithm can have false negatives, namely when
𝑝 ∈ 𝐼 has a nonzero 𝐺-reduced form, i. e., the reduction process reaches some 𝑓 ∈ 𝐼 that does
not have a monomial 𝑚 divisible by the leading monomial of some 𝑔 ∈ 𝐺.
A Gröbner basis 𝐺 for 𝐼 is a finite set of generators satisfying the additional constraint that
every nonzero element of 𝐼 has a monomial 𝑚 that is divisible by the leading monomial of some
element of 𝐺. In this case, the above algorithm for deciding membership in 𝐼 is always correct.
In fact, this gives another characterization of when a finite generating set 𝐺 is a Gröbner basis.
Yet another characterization is that every polynomial 𝑝 has a unique 𝐺-reduced form 𝑓 .
In the overview, we claimed that the set 𝐺 of polynomials EVC𝑙𝑘 [𝐶 t 𝐿] form a Gröbner basis
for Van[RFE𝑙𝑘 ], where 𝐶 ⊆ [𝑛] is a fixed core of size 𝑘 + 1 and 𝐿 ranges over the (𝑙 + 1)-subsets of
[𝑛] \ 𝐶. This holds with respect to any monomial ordering such that, for every 𝐿, 𝑥 𝐿 𝑖∈𝐿 𝑥 𝑖
Î
is the leading monomial of EVC𝑙𝑘 [𝐶 t 𝐿]. Examples of such orderings include all lexicographic
orderings where the variables outside 𝐶 have higher priority than the variables inside 𝐶.
Lemma 3.8 implies that every nonzero polynomial in Van[RFE𝑙𝑘 ] has a monomial with more
than 𝑙 variables outside of 𝐶, which is to say that the monomial is divisible by 𝑥 𝐿 for some
𝐿 ⊆ [𝑛] \ 𝐶 of size 𝑙 + 1. As 𝑥 𝐿 is the leading monomial of EVC𝑙𝑘 [𝐶 t 𝐿], we conclude that every
nonzero polynomial in Van[RFE𝑙𝑘 ] has a monomial that is divisible by the leading term of some
element of 𝐺, i. e., 𝐺 is a Gröbner basis.
One can interpret Proposition 3.6 as performing a reduction step of 𝑝 by 𝐺. Lemma 3.7
keeps performing this step until it is no longer possible, yielding a 𝐺-reduced form 𝑓 of 𝑝 that is
(𝑘 + 1, 𝑙)-cored with core 𝐶. Lemma 3.8 implies that any two (𝑘 + 1, 𝑙)-cored representatives
modulo 𝐼 of the same polynomial 𝑝 coincide, so every polynomial 𝑝 has a unique 𝐺-reduced
form. This is another way to see that the set 𝐺 is a Gröbner basis.
Instantiation for SV. By the connection between SV and RFE, a generating set for Van[RFE𝑙−1𝑙 ]
𝑙
induces a generating set for Van[SV ]. We provide an explicit expression as an instantiation of
Theorem 1.3 and Proposition 3.4.
Corollary 3.9. Let 𝑙, 𝑛 ∈ ℕ and let 𝑎 𝑖 for 𝑖 ∈ [𝑛] be distinct elements of 𝔽 . For any fixed set 𝐶 ⊆ [𝑛] of
size 𝑙, the polynomials EVCSV𝑙 [𝐶 t 𝐿] form a generating set of minimum size for Van[SV𝑙 ] when 𝐿
ranges over all (𝑙 + 1)-subsets of [𝑛] that are disjoint from 𝐶. Here, for any 𝑆 = {𝑖1 , . . . , 𝑖 2𝑙+1 } ⊆ [𝑛]
with 𝑖1 < · · · < 𝑖2𝑙+1 , Õ Ö
EVCSV𝑙 [𝑆] 0
𝛾𝑆\𝑇,𝑇 · 𝑥𝑖 ,
𝑇⊆𝑆 𝑖∈𝑇
|𝑇 |=𝑙+1
where
0 ©Ö Ö
𝛾𝑆\𝑇,𝑇 (−1)XInv(𝑆\𝑇,𝑇) · (𝑎 𝑖 − 𝑎 𝑗 )® · det(𝐴𝑆\𝑇 ) · det(𝐴𝑇 ).
ª
(3.6)
« 𝑖∈𝑇 𝑗∈[𝑛]\{𝑖} ¬
Proof. By Proposition 2.2, for any polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], 𝑝(SV𝑙 ) = 0 iff (𝑝 ◦ 𝐴)(RFE𝑙−1
𝑙 )=
𝑛 𝑛
0, where 𝐴 : 𝔽 → 𝔽 is the invertible transformation that divides each variable 𝑥 𝑖 by
𝑙−1
𝑗∈[𝑛]\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 ). Since the vanishing ideal of RFE𝑙
Î
coincides with the ideal generated by the
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polynomials EVC𝑙−1 𝑙
𝑙 [𝐶 t 𝐿], it follows that the vanishing ideal of SV coincides with the ideal
generated by the polynomials
EVC𝑙−1 −1
𝑙 [𝐶 t 𝐿] ◦ 𝐴 . (3.7)
For 𝑇 ⊆ 𝑆 𝐶 t 𝐿 with |𝑇 | = 𝑙 + 1, the coefficient of 𝑖∈𝑇 𝑥 𝑖 in EVC𝑙−1 𝛾𝑆\𝑇,𝑇 . By
Î
𝑙 [𝑆] is given by Î
the construction of 𝐴, 𝐴 multiplies 𝑥 𝑖 by 𝑗∈[𝑛]\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 ). Thus, the coefficient of 𝑖∈𝑇 𝑥 𝑖 in
−1 Î
(3.7) equals 𝛾𝑆\𝑇,𝑇
0
given by (3.6) and EVC𝑙−1
𝑙 [𝐶 t 𝐿] ◦ 𝐴
−1 = EVCSV 𝑙 [𝑆].
In comparison to the coefficients 𝛾𝑆\𝑇,𝑇 of the polynomial EVC𝑙𝑙−1 [𝑆], the coefficients 𝛾𝑆\𝑇,𝑇
0
of EVCSV𝑙 [𝑆] contain an additional term, namely the middle term on the right-hand side of (3.6).
As a consequence, each coefficient of EVCSV𝑙 [𝑆] depends on all abscissas 𝑎 1 , . . . , 𝑎 𝑛 , whereas the
coefficients of EVC𝑙𝑘 [𝑆] only depend on the abscissas with indices in 𝑆. This reflects a difference
in setup between the two generators: The substitution for a variable 𝑥 𝑖 is a multivariate function
of all abscissas in SV versus a univariate function of the abscissa 𝑎 𝑖 only in RFE. The difference
represents one reason why RFE is more convenient to work with than SV, even though both
have essentially the same power.
A more important reason is our derivation of the generating set EVC𝑙𝑘 in Lemma 3.1. Our
approach hinges on the fact that the substitutions for a variable 𝑥 𝑖 induce linear equations
involving the seed variables 𝑔 𝑘 , . . . , 𝑔0 , ℎ 𝑙 , . . . , ℎ 0 , with coefficients being expressions in terms
of the polynomial variables 𝑥 1 , . . . , 𝑥 𝑛 and abscissas 𝑎1 , . . . , 𝑎 𝑛 . Collecting 𝑘 + 𝑙 + 2 of such
equations yields as many linear constraints as unknowns, which suffices to derive a nontrivial
element of the vanishing ideal. The substitutions (1.1) for 𝑥 𝑖 made by SV𝑙 similarly induce linear
equations, though not between the mere seed variables 𝑦1 , 𝑧1 , . . . , 𝑦 𝑙 , 𝑧 𝑙 but between monomials
in the seed variables, namely the constant monomial and the monomials 𝑧 𝑡 𝑦𝑡𝑑 for 𝑡 ∈ [𝑙] and
𝑑 ∈ {0, . . . , 𝑛 − 1}. In contrast to the setting of RFE, even if we collect all of those equations,
namely 𝑛 linear equations in 𝑛𝑙 + 1 unknowns, this does not give us enough information to
derive a nontrivial element of the vanishing ideal.
4 Zoom Lemma
Throughout the paper we make repeated use of a key technical tool, the Zoom Lemma. The
lemma allows us to zoom in on the contributions of the monomials in a polynomial 𝑝 that have
prescribed degrees in a subset of the variables. We introduce the following terminology for
prescribing degrees.
Definition 4.1 (degree pattern). Let 𝐽 ⊆ [𝑛]. A degree pattern with domain 𝐽 is a 𝐽-indexed tuple
𝑑 ∈ ℕ 𝐽 of nonnegative integers. A degree pattern 𝑑 matches a monomial 𝑚 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] if, for
every 𝑗 ∈ 𝐽, 𝑚 has degree exactly 𝑑 𝑗 in 𝑥 𝑗 . We say that 𝑑 is in 𝑝 if 𝑑 matches some monomial in 𝑝.
For any fixed 𝐽, every polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] can be written uniquely in the form
Õ
𝑝= 𝑝𝑑 · 𝑥 𝑑
𝑑∈ℕ 𝐽
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
𝑑𝑗
where 𝑥 𝑑 𝑗∈𝐽 𝑥 𝑗 and 𝑝 𝑑 depends only on variables not indexed by 𝐽. We refer to 𝑝 𝑑 as the
Î
coefficient of 𝑥 𝑑 in 𝑝.
The notation 𝑝 𝑑 can be viewed as a generalization of the common one for the coefficient of
degree 𝑑 of a univariate polynomial 𝑝.
Our technique allows us to zoom in on the contributions of the coefficients 𝑝 𝑑 of degree
patterns 𝑑 that satisfy the following additional constraint.
Definition 4.2 (extremal degree pattern). Let 𝐾, 𝐿 ⊆ [𝑛]. A degree pattern 𝑑∗ ∈ ℕ 𝐾∪𝐿 is
(𝐾, 𝐿)-extremal in a polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] if 𝑑 ∗ is the unique degree pattern 𝑑 ∈ ℕ 𝐾∪𝐿 in 𝑝
that satisfies both
(i) 𝑑 𝑗 ≤ 𝑑∗𝑗 for all 𝑗 ∈ 𝐾, and
(ii) 𝑑 𝑗 ≥ 𝑑∗𝑗 for all 𝑗 ∈ 𝐿.
The notion of extremality in Definition 4.2 is closely related to standard notions of minimality
and maximality of tuples of numbers. A 𝐽-tuple 𝑑 ∗ is minimal in a set 𝐷 of such tuples if the only
tuple 𝑑 ∈ 𝐷 that satisfies 𝑑 𝑗 ≤ 𝑑 ∗𝑗 for all 𝑗 ∈ 𝐽, is 𝑑 ∗ itself. A maximal tuple is defined similarly by
replacing ≤ by ≥. Minimality is equivalently (𝐽, ∅)-extremality, and maximality is equivalently
(∅, 𝐽)-extremality.
When 𝐾 and 𝐿 intersect, note that only degree patterns 𝑑 ∈ ℕ 𝐾∪𝐿 with 𝑑 𝑗 = 𝑑 ∗𝑗 for all 𝑗 ∈ 𝐾 ∩ 𝐿
affect whether 𝑑 ∗ is (𝐾, 𝐿)-extremal.
The above terminology lets us state our key technical lemma succinctly.
Lemma 4.3 (Zoom Lemma). Let 𝐾, 𝐿 ⊆ [𝑛], let 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], and let 𝑑 ∗ ∈ ℕ 𝐾∪𝐿 be a degree
pattern that is (𝐾, 𝐿)-extremal in 𝑝. If the coefficient 𝑝 𝑑∗ is nonzero upon the substitution
𝑗∈𝐾\𝐿 (𝑎 𝑖 − 𝑎 𝑗 )
Î
𝑥𝑖 ← 𝑧 · Î ∀𝑖 ∈ [𝑛] \ (𝐾 ∪ 𝐿) (4.1)
𝑗∈𝐿\𝐾 (𝑎 𝑖 − 𝑎 𝑗 )
where 𝑧 is a fresh variable, then RFE𝑙𝑘 hits 𝑝 for any 𝑘 ≥ |𝐾| and 𝑙 ≥ |𝐿|.
Note that the result of substituting (4.1) into 𝑝 𝑑∗ is a univariate polynomial 𝑞 in 𝑧. In the case
where 𝑝 is homogeneous, 𝑞 has a single monomial, so 𝑞 is nonzero iff 𝑞 is nonzero at 𝑧 = 1.
In general, it suffices for 𝑞 to be nonzero at some point 𝑧 ∈ 𝔽 . As for the conclusion, the most
interesting settings in Lemma 4.3 are 𝑘 = |𝐾| and 𝑙 = |𝐿|. This is because the range of RFE𝑙𝑘 is
contained in the range of RFE𝑙𝑘0 for 𝑘 0 ≥ 𝑘 and 𝑙 0 ≥ 𝑙. Also, whereas many or our instantiations
0
of the Zoom Lemma have 𝐾 and 𝐿 disjoint, this is not necessary for the lemma to hold.1
Let us first see how the Zoom Lemma allows us to complete the proof of Theorem 1.3. There
are several ways to do so; we present a fairly generic way.
1In fact, allowing 𝐾 and 𝐿 to overlap is useful in Section 5 (see Proposition 5.10) and Section 8 (see Proposition 8.8).
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Proof of Lemma 3.8 from the Zoom Lemma. Let 𝐶 ⊆ [𝑛] denote a core of size at most 𝑘 + 1 for 𝑟.
We construct subsets 𝐾, 𝐿 ⊆ [𝑛] with |𝐾| ≤ 𝑘 and |𝐿| ≤ 𝑙, and a degree pattern 𝑑 ∗ with domain
𝐾 ∪ 𝐿 that is (𝐾, 𝐿)-extremal in 𝑟 such that 𝑟 𝑑∗ is nonzero upon the substitution (4.1). The Zoom
Lemma then implies that RFE𝑙𝑘 hits 𝑟.
The construction consists of two steps. First, we pick 𝑖 ∗ ∈ 𝐶 arbitrarily. (We can assume
without loss of generality that 𝐶 is nonempty because if 𝐶 is a core, then so is 𝐶 with an
additional element.) We also set 𝐾 𝐶 \ {𝑖 ∗ }, and let 𝑑+ be a degree pattern with domain 𝐾 that
matches a monomial in 𝑟 and that is minimal among all such degree patterns. The existence
of 𝑑+ follows from the fact that 𝑟 is nonzero. By construction, |𝐾| ≤ (𝑘 + 1) − 1 = 𝑘 and 𝑟 𝑑+ is
nonzero.
Second, we pick a degree pattern 𝑑− with domain [𝑛] \ 𝐶 that matches a monomial in 𝑟 𝑑+
and that is maximal among all such degree patterns. The existence of 𝑑− follows from the fact
that 𝑟 𝑑+ is nonzero. Let 𝐿 denote the set of indices 𝑗 ∈ [𝑛] \ 𝐶 on which 𝑑− is positive. The
hypothesis that 𝐶 is a (𝑘 + 1, 𝑙)-core for 𝑟 implies that |𝐿| ≤ 𝑙. By construction, the restriction of
𝑑− to the domain 𝐿 is maximal among the degree patterns with domain 𝐿 in 𝑟 𝑑+ .
Note that 𝐾 and 𝐿 are disjoint, because 𝐾 ⊆ 𝐶 and 𝐿 ⊆ [𝑛] \ 𝐶. We define 𝑑 ∗ as the degree
pattern with domain 𝐾 t 𝐿 that agrees with 𝑑+ on 𝐾 and with 𝑑− on 𝐿. The minimality and
maximality properties of 𝑑+ and 𝑑− imply that 𝑑 ∗ is (𝐾, 𝐿)-extremal in 𝑟. As there is at least
one monomial in 𝑟 that agrees with the degree pattern 𝑑 ∗ , the coefficient 𝑟 𝑑∗ is nonzero. Since
𝐾 includes all of 𝐶 but 𝑖 ∗ , 𝑟 𝑑∗ cannot depend on variables indexed by 𝐶 other than 𝑥 𝑖 ∗ . By the
maximality of 𝑑− on [𝑛] \ 𝐶 and the fact that 𝐿 contains all indices in [𝑛] \ 𝐶 on which 𝑑− is
positive, 𝑟 𝑑∗ cannot depend on any variable in [𝑛] \ 𝐶. Thus, 𝑟 𝑑∗ is a nonzero polynomial that
depends only on 𝑥 𝑖 ∗ . It follows that substituting (4.1) into 𝑟 𝑑∗ yields a nonzero polynomial in
𝑧.
Before giving a formal proof of the Zoom Lemma, we provide some intuition for the
mechanism behind it, and we explain how the choice of the substitution (4.1) and the extremality
requirement arise. We consider 𝑘 = |𝐾| and 𝑙 = |𝐿|, and focus on the setting of homogeneous
polynomials 𝑝, in which case we can set 𝑧 = 1 without loss of generality.
We start with the special case where (i) ℓ = 0, or equivalently 𝐿 = ∅, and (ii) the degree
pattern 𝑑 ∗ ∈ ℕ 𝐾 is zero in every coordinate, so 𝑥 𝑑 is the constant monomial 1. We can zoom
∗
in on 𝑝 𝑑∗ by setting all variables 𝑥 𝑗 for 𝑗 ∈ 𝐾 to zero. The generator RFE0𝑘 allows us to do so by
picking a seed 𝑓 such that 𝑓 (𝑎 𝑗 ) = 0 for all 𝑗 ∈ 𝐾, namely
Ö
𝑓 (𝛼) (𝛼 − 𝑎 𝑗 ). (4.2)
𝑗∈𝐾
The evaluation of 𝑝 at RFE( 𝑓 ) coincides with the evaluation of 𝑝 𝑑∗ at RFE( 𝑓 ), which is precisely
(4.1) with 𝑧 = 1. If the evaluation is nonzero, then evidently RFE0𝑘 hits 𝑝, as desired.
In order to handle more general degree patterns 𝑑 ∗ ∈ ℕ 𝐾 , we introduce a fresh parameter 𝜉 𝑗
for each 𝑗 ∈ 𝐾, and replace 𝑎 𝑗 in (4.2) by 𝑎 𝑗 − 𝜉 𝑗 , i. e., we consider the seeds
Ö
𝑓ˆ(𝛼) (𝛼 − 𝑎 𝑗 + 𝜉 𝑗 ), (4.3)
𝑗∈𝐾
T HEORY OF C OMPUTING, Volume 20 (1), 2024, pp. 1–70 25
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where the hat indicates a dependency on the fresh parameters. For each 𝑖, 𝑓ˆ(𝑎 𝑖 ) is a multivariate
polynomial in 𝜉 𝑗 , 𝑗 ∈ 𝐾, and RFE( 𝑓ˆ) applies the substitution 𝑥 𝑖 ← 𝑓ˆ(𝑎 𝑖 ) for each 𝑖 ∈ [𝑛]. The
critical property is that 𝑓ˆ(𝑎 𝑖 ) contains the factor 𝜉𝑖 for 𝑖 ∈ 𝐾 but not for 𝑖 ∉ 𝐾. More precisely,
(
𝑐ˆ 𝑖 · 𝜉𝑖 𝑖∈𝐾
𝑓ˆ(𝑎 𝑖 ) = ,
𝑐ˆ 𝑖 𝑖∉𝐾
where 𝑐ˆ 𝑖 𝑗∈𝐾\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 + 𝜉 𝑗 ) is a multivariate polynomial in the parameters 𝜉 with nonzero
Î
constant term, namely 𝑐 𝑖 𝑗∈𝐾\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 ).
Î
For any monomial 𝑚 with matching degree pattern 𝑑 ∈ ℕ 𝐾 , we have
𝑚(RFE( 𝑓ˆ)) = 𝑚(𝑐ˆ) · 𝜉 𝑑 = 𝑚 𝑑 (𝑐ˆ) · 𝑐ˆ 𝑑 · 𝜉 𝑑 .
Here we see that, when 𝑚(RFE( 𝑓ˆ)) is expanded as a linear combination of monomials in the 𝜉 𝑗 ,
the combination contains only monomials divisible by 𝜉 𝑑 and the coefficient of 𝜉 𝑑 is nonzero
(namely 𝑐 𝑑 ).
In the expansion of 𝑝(RFE( 𝑓ˆ)), the coefficient of 𝜉 𝑑
∗
(a) has a contribution 𝑚(𝑐) = 𝑚 𝑑∗ (𝑐) · 𝑐 𝑑 from each monomial 𝑚 in 𝑝 that matches 𝑑 ∗ , and
∗
(b) may have contributions from other monomials 𝑚 in 𝑝 but only from those whose degree
pattern on 𝐾 is smaller than 𝑑 ∗ , i. e., only if deg 𝑗 (𝑚) ≤ 𝑑 ∗𝑗 for all 𝑗 ∈ 𝐾.
By adding the contributions of all monomials 𝑚 with degree pattern 𝑑 ∗ we obtain
𝑝 𝑑∗ (RFE( 𝑓ˆ)) · RFE( 𝑓ˆ)𝑑 = 𝑝 𝑑∗ ( 𝑐ˆ) · 𝑐ˆ 𝑑 · 𝜉 𝑑 .
∗ ∗ ∗
By properties (a) and (b) above, we conclude that the coefficient of the monomial 𝜉 𝑑 in 𝑝(RFE( 𝑓ˆ)):
∗
(a’) has a contribution of 𝑝 𝑑∗ (𝑐) · 𝑐 𝑑 from the monomials matching 𝑑 ∗ , and
∗
(b’) cannot have any additional contributions provided that there are no degree patterns on 𝐾
in 𝑝 that are smaller than 𝑑 ∗ .
For a degree pattern 𝑑 ∗ in 𝑝, condition (b’) can be formulated as the minimality of 𝑑 ∗ among the
degree patterns on 𝐾 in 𝑝, which is exactly the requirement that 𝑑 ∗ is (𝐾, 𝐿)-extremal in 𝑝 for
𝐿 = ∅. Under this condition we conclude that the coefficient of the monomial 𝜉 𝑑 in 𝑝(RFE( 𝑓ˆ))
∗
equals 𝑝 𝑑∗ (𝑐) · 𝑐 𝑑 . Note that 𝑐 𝑑 is nonzero. Since 𝑝 𝑑∗ (𝑐) only depends on the components 𝑐 𝑖 for
∗ ∗
𝑖 ∈ [𝑛] \ 𝐾, and those components agree with (4.1) for 𝑧 = 1, the coefficient of the monomial 𝜉 𝑑
∗
in 𝑝(RFE( 𝑓ˆ)) is nonzero if and only if 𝑝 𝑑∗ is nonzero at the point (4.1) with 𝑧 = 1. Thus, for a
homogeneous polynomial 𝑝, the hypotheses of the lemma imply that 𝑝(RFE( 𝑓ˆ)) is a nonzero
polynomial in the parameters 𝜉. It follows that a random setting of the parameters 𝜉 yields a
seed 𝑓 0 for RFE0𝑘 such that 𝑝(RFE( 𝑓 0)) is nonzero. This shows that RFE0𝑘 hits 𝑝.
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The symmetric case 𝑘 = 0 can be obtained from the case 𝑙 = 0 by transforming 𝑥 𝑖 ↦→ 𝑥 −1 𝑖
for
˜ 𝑙
each 𝑖 ∈ [𝑛]. The transformation maps a seed 𝑓 for RFE𝑙 into a seed 𝑓 for RFE0 , wherein the
0
zeroes of 𝑓˜ come from the poles of 𝑓 . Given a polynomial 𝑝(𝑥1 , . . . , 𝑥 𝑛 ), we similarly transform
the variables and clear denominators to obtain the polynomial 𝑝(𝑥 ˜ 1 , . . . , 𝑥 𝑛 ) 𝑝(𝑥1−1 , . . . , 𝑥 𝑛−1 )·𝑥 𝑔 ,
where 𝑔 is any degree pattern with domain [𝑛] for which 𝑔 𝑖 is at least the degree of 𝑥 𝑖 in 𝑝 for
every 𝑖 ∈ [𝑛]. We apply the previous case of the Zoom Lemma to 𝑝˜ and obtain the new case
of the Zoom Lemma for 𝑝. Note that a monomial with degree pattern 𝑑˜ in 𝑝˜ corresponds to a
monomial with degree pattern 𝑑 = 𝑔 − 𝑑˜ in 𝑝. It follows that 𝑑e∗ is minimal in 𝑝˜ iff 𝑑 ∗ is maximal
in 𝑝, which is exactly the (𝐾, 𝐿)-extremality requirement of the Zoom Lemma in the case where
𝐾 = ∅.
The above arguments for the special cases 𝑙 = 0 and 𝑘 = 0 carry through for arbitrary
polynomials 𝑝 under the assumption that 𝑝 𝑑∗ is nonzero upon the substitution (4.1) with 𝑧 = 1,
i. e., that the univariate polynomial 𝑞(𝑧) obtained by substituting (4.1) in 𝑝 𝑑∗ is nonzero at 𝑧 = 1.
The homogeneity of 𝑝 was only used to conclude that if 𝑞 is nonzero, then 𝑞 is nonzero at 𝑧 = 1.
To handle polynomials 𝑝 where 𝑞 may be nonzero but zero at 𝑧 = 1, we run the above argument
with an arbitrary value of 𝑧 ∈ 𝔽 where 𝑞 is nonzero. We can do so by including an additional
factor of 𝑧 on the right-hand sides of (4.2) and (4.3), i. e., by considering 𝑓 (𝛼) 𝑧 · 𝑗∈𝐾 (𝛼 − 𝑎 𝑗 )
Î
and 𝑓ˆ(𝛼) 𝑧 · (𝛼 − 𝑎 𝑗 + 𝜉 𝑗 ), respectively. Both expressions correspond to valid seeds for
Î
𝑗∈𝐾
RFE0𝑘 in the roots parametrization.
The case for general 𝑘 and 𝑙 follows in a similar fashion, introducing parameters for the
zeroes as well as the poles of the seed 𝑓 , considering the monomial in those parameters with
degree pattern determined by 𝑑 ∗ , and clearing denominators.
Proof of Zoom Lemma. Let 𝐾, 𝐿, 𝑝, and 𝑑 ∗ be as in the lemma statement. Fix 𝑧 to a value in 𝔽
such that 𝑝 𝑑∗ is nonzero upon the substitution (4.1). Such a value exists by the hypothesis of
the lemma (for large enough 𝔽 ). Since the range of RFE𝑙𝑘 is contained in the range of RFE𝑙𝑘0 for
0
𝑘 0 ≥ 𝑘 and 𝑙 0 ≥ 𝑙, it suffices to show that RFE𝑙𝑘 hits 𝑝 for 𝑘 = |𝐾| and 𝑙 = |𝐿|. Let 𝜉 𝑗 for each 𝑗 ∈ 𝐾
and 𝜂 𝑗 for each 𝑗 ∈ 𝐿 be fresh indeterminates. We denote by 𝔽b the field of rational functions in
those indeterminates with coefficients in 𝔽 , and by 𝑉 the subset of elements that, when written
in lowest terms, have denominators with nonzero constant terms. Let Φ : 𝑉 → 𝔽 map each
element of 𝑉 to the result of substituting 𝜉 𝑗 ← 0 for each 𝑗 ∈ 𝐾 and 𝜂 𝑗 ← 0 for each 𝑗 ∈ 𝐿. The
result is always well-defined.
Define 𝑓ˆ ∈ 𝔽b(𝛼) as follows:
(𝛼 − 𝑎 𝑗 + 𝜉 𝑗 )
Î
𝑗∈𝐾
𝑓ˆ(𝛼) 𝑧 · Î .
𝑗∈𝐿 (𝛼 − 𝑎 𝑗 + 𝜂 𝑗 )
The substitution RFE( 𝑓ˆ) effects 𝑥 𝑖 ← 𝑓ˆ(𝑎 𝑖 ) ∈ 𝔽b for each 𝑖 ∈ [𝑛]. We claim that 𝑝(RFE( 𝑓ˆ)) is
nonzero. This suffices to conclude that RFE𝑙𝑘 hits 𝑝, because substituting 𝜉 𝑗 and 𝜂 𝑗 by a random
scalar from 𝔽 transforms 𝑓ˆ into a seed 𝑓 0 such that, with high probability, 𝑓 0 is a valid seed for
RFE𝑙𝑘 and 𝑝(RFE( 𝑓 0)) ≠ 0. Henceforth we show that 𝑝(RFE( 𝑓ˆ)) ≠ 0.
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For each 𝑖 ∈ [𝑛], there exists 𝑐ˆ 𝑖 ∈ 𝑉 with Φ( 𝑐ˆ 𝑖 ) ≠ 0 such that
𝑐ˆ 𝑖 · 𝜂𝜉𝑖𝑖 𝑖 ∈ 𝐾∩𝐿
𝑐 𝑖 · 𝜉𝑖 𝑖 ∈ 𝐾\𝐿
ˆ
𝑓ˆ(𝑎 𝑖 ) =
, (4.4)
𝑐ˆ𝑖 · 𝜂𝑖 𝑖 ∈ 𝐿\𝐾
1
𝑐ˆ
𝑖 ∉𝐾∪𝐿
𝑖
namely
𝑗∈𝐾\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 + 𝜉 𝑗 )
Î
𝑐ˆ 𝑖 = 𝑧 · Î .
𝑗∈𝐿\{𝑖} (𝑎 𝑖 − 𝑎 𝑗 + 𝜂 𝑗 )
For 𝑖 ∉ 𝐾 ∪ 𝐿, Φ( 𝑐ˆ 𝑖 ) is moreover the value substituted into 𝑥 𝑖 by (4.1).
Let 𝐷 denote the set of all degree patterns 𝑑 ∈ ℕ 𝐾∪𝐿 that match a monomial in 𝑝. We have
that Õ
𝑝= 𝑝𝑑 · 𝑥 𝑑 . (4.5)
𝑑∈𝐷
For 𝑑 ∈ 𝐷, define 𝑞ˆ 𝑑 to be the result of substituting 𝑥 𝑖 ← 𝑐ˆ 𝑖 into 𝑝 𝑑 for each 𝑖 ∈ [𝑛].
Combining (4.4) and (4.5), we obtain
Õ 𝜉 𝑑| 𝐾
𝑝(RFE( 𝑓ˆ)) = 𝑞ˆ 𝑑 · 𝑐ˆ 𝑑 · , (4.6)
𝑑∈𝐷
𝜂 𝑑| 𝐿
where 𝑑| 𝐾 and 𝑑| 𝐿 respectively are the restrictions of 𝑑 onto the domains 𝐾 and 𝐿 respectively.
Fix any function 𝜓 : [𝑘 + 𝑙] → 𝐾 ∪ 𝐿 such that 𝜓 establishes a bijection between {1, . . . , 𝑘}
and 𝐾 and establishes a bijection between {𝑘 + 1, . . . , 𝑘 + 𝑙} and 𝐿. For 𝑗 ∈ {1, . . . , 𝑘}, let 𝜁 𝑗 be
an alias for 𝜉𝜓(𝑗) , and for 𝑗 ∈ {𝑘 + 1, . . . , 𝑘 + 𝑙}, let 𝜁 𝑗 be an alias for 𝜂𝜓(𝑗) . For each 𝑑 ∈ ℕ 𝐾∪𝐿 ,
define a corresponding 𝛿 ∈ ℤ 𝑘+𝑙 given by 𝛿 𝑗 = 𝑑𝜓(𝑗) for 𝑗 ∈ {1, . . . , 𝑘} and 𝛿 𝑗 = −𝑑𝜓(𝑗) for
𝑗 ∈ {𝑘 + 1, . . . , 𝑘 + 𝑙}. Let Δ ⊆ ℤ 𝑘+𝑙 consist of the 𝛿 corresponding to each 𝑑 ∈ 𝐷. Finally, for
each 𝑑 ∈ 𝐷 with corresponding 𝛿 ∈ Δ, define 𝑐ˆ 𝛿 𝑞ˆ 𝑑 · 𝑐ˆ 𝑑 , capturing the first two factors in the
𝑑-th term of (4.6). Rewritten in this notation, (4.6) becomes
𝑘+𝑙
𝛿𝑗
Õ Ö
𝑐ˆ 𝛿 · 𝜁𝑗 . (4.7)
𝛿∈Δ 𝑗=1
Our hypothesis that 𝑑 ∗ is (𝐾, 𝐿)-extremal in 𝑝 says that the only 𝑑 ∈ 𝐷 such that 𝑑 𝑗 ≤ 𝑑 ∗𝑗 for
every 𝑗 ∈ 𝐾 and 𝑑 𝑗 ≥ 𝑑 ∗𝑗 for every 𝑗 ∈ 𝐿, is 𝑑 = 𝑑 ∗ . Translated into a condition on the element
𝛿∗ ∈ Δ corresponding to 𝑑 ∗ , the hypothesis says that 𝛿∗ is minimal in Δ. Our other hypothesis
states that 𝑝 𝑑∗ does not vanish upon substituting (4.1). As (4.1) equates to substituting 𝑥 𝑖 ← Φ( 𝑐ˆ 𝑖 )
for 𝑖 ∉ 𝐾 ∪ 𝐿, this hypothesis equivalently states that Φ( 𝑞ˆ 𝑑∗ ) is nonzero. Since for each 𝑗 ∈ 𝐾 ∪ 𝐿
we have Φ( 𝑐ˆ 𝑗 ) ≠ 0, we conclude that Φ( 𝑐ˆ 𝛿∗ ) ≠ 0. That 𝑝(RFE( 𝑓ˆ)) is nonzero now follows from
the next proposition.
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Proposition 4.4. Let 𝔽b = 𝔽 (𝜁 1 , . . . , 𝜁 𝑟 ) be the field of rational functions in indeterminates 𝜁1 , . . . , 𝜁 𝑟 ,
let 𝑉 ⊆ 𝔽b consist of the rational functions whose denominator has nonzero constant term, and let
Φ : 𝑉 → 𝔽 be the function that maps each rational function in 𝑉 to its value after substituting 𝜁 𝑗 ← 0
for all 𝑗 ∈ [𝑟]. Let
𝑟
𝛿𝑗
Õ Ö
𝑠= 𝑐ˆ 𝛿 · 𝜁𝑗
𝛿∈Δ 𝑗=1
where Δ ⊆ ℤ𝑟 is some finite set, and we have 𝑐ˆ 𝛿 ∈ 𝑉 for every 𝛿 ∈ Δ. If there exists 𝛿∗ ∈ Δ that is
minimal in Δ and for which Φ( 𝑐ˆ 𝛿∗ ) ≠ 0, then 𝑠 ≠ 0.
Proof. By clearing denominators, we may assume without loss of generality that, for every 𝛿 ∈ Δ
and every 𝑗 ∈ [𝑟], 𝛿 𝑗 ≥ 0, and that, for every 𝛿 ∈ Δ, 𝑐ˆ 𝛿 is a polynomial in 𝜁 1 , . . . , 𝜁 𝑟 . In this case,
all quantities in the sum for 𝑠 are polynomials in 𝜁1 , . . . , 𝜁 𝑟 . The minimality hypothesis on 𝛿∗
implies that the coefficient of 𝑟𝑗=1 𝜁 𝛿𝑗 𝑗 in the monomial expansion of 𝑠 is precisely the constant
Î ∗
coefficient of 𝑐ˆ 𝛿∗ , and the hypothesis Φ(𝑐ˆ 𝛿∗ ) ≠ 0 asserts that this coefficient is nonzero.
5 Membership Test
In this section we develop the structured membership test for the vanishing ideal Van[RFE𝑙𝑘 ]
given in Theorem 1.8. We begin with some basic results regarding membership to Van[RFE𝑙𝑘 ]
and then develop a criterion for multilinear polynomials.
Basic properties. It is well-known that Van[SV𝑙 ] does not contain any polynomial with a
monomial of support at most 𝑙, i. e., a monomial involving at most 𝑙 variables. We generalize
the lower bound on the support to Van[RFE𝑙𝑘 ] and also establish an upper bound in the case of
multilinear polynomials. Note that for multilinear monomials support conditions translate into
degree conditions.
Proposition 5.1. If a polynomial 𝑝 contains a monomial of support at most 𝑙, then RFE𝑙𝑘 hits 𝑝. If a
multilinear polynomial 𝑝 in the variables 𝑥 1 , . . . , 𝑥 𝑛 contains a monomial of support at least 𝑛 − 𝑘, then
RFE𝑙𝑘 hits 𝑝.
The known proofs of the first part for SV𝑙 make use of partial derivatives. We establish the
generalization for RFE𝑙𝑘 using our generating set in Theorem 1.3, whose analysis hinges on the
Zoom Lemma. A similar argument works for the second part, but we opt to establish it via a
black-box reduction to the first part for multilinear polynomials. The approach illustrates the
utility of our generalization of SV𝑙 to RFE𝑙𝑘 since even for SV𝑙 we need to consider settings of the
parameters 𝑘 and 𝑙 other than 𝑘 = 𝑙 − 1.
Proof. For the first part, by Proposition 3.4 none of the polynomials EVC𝑙𝑘 contain a monomial
of support 𝑙 or less. The same holds for the nonzero polynomials in the ideal generated by
these polynomials, which by Theorem 1.3 equals Van[RFE𝑙𝑘 ]. Thus, every polynomial that has a
monomial of support at most 𝑙, is hit by RFE𝑙𝑘 .
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For the second part, consider 𝑞(𝑥 1 , . . . , 𝑥 𝑛 ) 𝑥 1 · · · · · 𝑥 𝑛 · 𝑝(1/𝑥 1 , . . . , 1/𝑥 𝑛 ). Note that if 𝑝
is a multilinear polynomial in the variables 𝑥 1 , . . . , 𝑥 𝑛 , then so is 𝑞. If a multilinear 𝑝 has a
monomial of support at least 𝑛 − 𝑘, then 𝑞 has a monomial of support at most 𝑘. By the first
part, RFE𝑙𝑘 hits 𝑞. Since the mapping 𝑥 𝑖 ← 1/𝑥 𝑖 transforms RFE𝑙𝑘 into RFE𝑙𝑘 , we conclude that
RFE𝑙𝑘 hits 𝑝.
Another feature of SV that generalizes to RFE is that the generator separates the homogeneous
components of a given polynomial 𝑝. The feature allows us to reduce the general case of testing
membership in Van[RFE𝑙𝑘 ] to the homogeneous case, as was already effectively used in the proof
of the Zoom Lemma.
Proposition 5.2. For any polynomial 𝑝, 𝑝 vanishes upon substituting RFE if and only if every
homogeneous component of 𝑝 vanishes upon substituting RFE.
Proof. For any seed 𝑓 for RFE and any scalar 𝑧, the rescaled substitution 𝑧 · RFE( 𝑓 ) is in the
range of RFE, namely as RFE(𝑧 · 𝑓 ). It follows (provided that 𝔽 is sufficiently large) that 𝑝(RFE)
vanishes if and only if 𝑝(𝜁 · RFE) vanishes, where 𝜁 is a fresh indeterminate. We now consider
the expansion of 𝑝(𝜁 · RFE) as a polynomial in 𝜁. With 𝑝 (𝑑) as the degree-𝑑 homogeneous
component of 𝑝, we have
Õ Õ
𝑝(𝜁 · RFE) = 𝑝 (𝑑) (𝜁 · RFE) = 𝜁 𝑑 · 𝑝 (𝑑) (RFE).
𝑑 𝑑
The coefficient of 𝜁 𝑑 , 𝑝 (𝑑) (RFE), has no dependence on 𝜁. We deduce that 𝑝(𝜁 · RFE) is the zero
polynomial if and only if 𝑝 (𝑑) (RFE) vanishes for every 𝑑.
Criterion for multilinear polynomials. We now develop the full membership test for multilin-
ear polynomials given in Theorem 1.8. Condition 2 in Theorem 1.8 is closely related to the Zoom
Lemma. Note that for multilinear polynomials and disjoint 𝐾 and 𝐿, 𝜕𝐿 𝑝| 𝐾←0 coincides with
the coefficient 𝑝 𝑑∗ where 𝑑 ∗ is the degree pattern with domain 𝐾 t 𝐿, 0 in the positions of 𝐾, and
1 in the positions of 𝐿. Moreover, since 𝑝 is multilinear, the condition that 𝑑∗ be (𝐾, 𝐿)-extremal
in 𝑝 is automatically satisfied: The only multilinear monomial 𝑚 with support in 𝐾 t 𝐿 with
deg𝑥 𝑖 (𝑚) ≤ 𝑑 ∗𝑖 = 0 for all 𝑖 ∈ 𝐾 and deg𝑥 𝑖 (𝑚) ≥ 𝑑 ∗𝑖 = 1 for all 𝑖 ∈ 𝐿 is 𝑚 = 𝑥 𝑑 . This leads to the
∗
following specialization of the Zoom Lemma for multilinear polynomials with disjoint 𝐾 and 𝐿.
Lemma 5.3 (Zoom Lemma for multilinear polynomials). Let 𝐾, 𝐿 ⊆ [𝑛] be disjoint, and let
𝑝 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] be a multilinear polynomial. If 𝜕𝐿 𝑝| 𝐾←0 is nonzero upon the substitution
𝑗∈𝐾 (𝑎 𝑖 − 𝑎 𝑗 )
Î
𝑥𝑖 ← 𝑧 · Î ∀𝑖 ∈ [𝑛] \ (𝐾 t 𝐿), (5.1)
𝑗∈𝐿 (𝑎 𝑖 − 𝑎 𝑗 )
where 𝑧 is a fresh variable, then RFE𝑙𝑘 hits 𝑝 for any 𝑘 ≥ |𝐾| and 𝑙 ≥ |𝐿|.
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Observe that the substitution (5.1) in Lemma 5.3 coincides with (1.7) in Theorem 1.8. Thus,
for a multilinear polynomial 𝑝, condition 2 in Theorem 1.8 expresses that there is no way to
show that RFE𝑙𝑘 hits 𝑝 via an application of Lemma 5.3. The necessity of this condition for
membership of 𝑝 in Van[RFE𝑙𝑘 ] is clear. The necessity of Condition 1 in Theorem 1.8 is just the
special case of Proposition 5.1 for multilinear polynomials.
What remains to argue is that the combination of condition 1 and condition 2 is sufficient for
membership of 𝑝 in Van[RFE𝑙𝑘 ]. Equivalently, it remains to argue the following property for
every multilinear polynomial 𝑝 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] that only contain monomials of degrees between
𝑙 + 1 and 𝑛 − 𝑘 − 1: If 𝑝 is hit by RFE𝑙𝑘 then there is an application of Lemma 5.3 that exhibits
this fact. We actually prove the property for every multilinear 𝑝 of degree 𝑑 with 𝑙 ≤ 𝑑 ≤ 𝑛 − 𝑘.
We do so with a two-step strategy similar to one we used for the part of Theorem 1.3 that RFE𝑙𝑘
hits every polynomial outside of the ideal generated by instantiations of EVC𝑙𝑘 :
1. Modulo the ideal 𝐼 generated by a certain subset of the instantiations of EVC𝑙𝑘 , 𝑝 is equal
to a cored polynomial 𝑟 with certain parameters. Since 𝑝 ∉ Van[RFE𝑙𝑘 ] and 𝐼 ⊆ Van[RFE𝑙𝑘 ],
𝑟 needs to be nonzero.
2. For every such cored polynomial 𝑟 that is nonzero, we can apply Lemma 5.3 to prove that
RFE𝑙𝑘 hits 𝑟, i. e., condition 2 fails for 𝑟.
By linearity and the necessity of condition 2 for multilinear polynomials in Van[RFE𝑙𝑘 ]), we
conclude that the condition fails for 𝑝, as well.
The crux for the first step in the context of Theorem 1.3 is the transformation in Proposition 3.6,
which gradually gets closer to a cored polynomial with the desired parameters. In general, the
transformation in Proposition 3.6 does not maintain multilinearity. We show how to tweak the
transformation and preserve multilinearity at the expense of an increase in the size of the core.
Proposition 5.4. Let 𝑘, 𝑙, 𝑛, 𝑑 ∈ ℕ , let 𝐶 be a (𝑘 + 𝑑 − 𝑙)-subset of [𝑛], and let 𝐼 denote the ideal
generated by the polynomials EVC𝑙𝑘 [𝐾 t 𝐿] where 𝐾 ranges over all (𝑘 + 1)-subsets of 𝐶 and 𝐿 ranges
over all (𝑙 + 1)-subsets of [𝑛] \ 𝐾. Consider a multilinear monomial 𝑚 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] of degree at most
𝑑 such that |supp(𝑚) \ 𝐶 | > 𝑙. Modulo 𝐼, 𝑚 is equal to a linear combination of multilinear monomials
of the same degree as 𝑚 but whose non-𝐶-parts have lower degree than the non-𝐶-part of 𝑚.
Proof. Consider the subset 𝐿 ⊆ supp(𝑚) \ 𝐶 of size 𝑙 + 1 in the proof of Proposition 3.6, and
𝑥 𝐿 𝑖∈𝐿 𝑥 𝑖 . Since 𝑚 is multilinear, so is 𝑚 0 𝑚/𝑥 𝐿 , and |supp(𝑚 0)| ≤ 𝑑 − |𝐿| = 𝑑 − 𝑙 − 1.
Î
Provided |𝐶| ≥ (𝑘 + 1) + (𝑑 − 𝑙 − 1) = 𝑘 + 𝑑 − 𝑙, there exists a subset 𝐾 ⊆ 𝐶 of size 𝑘 + 1 that is
disjoint from supp(𝑚 0). We substitute EVC𝑙𝑘 [𝐶 t 𝐿] by EVC𝑙𝑘 [𝐾 t 𝐿] in the proof. EVC𝑙𝑘 [𝐾 t 𝐿] is
homogeneous and multilinear. By construction 𝐾 t 𝐿 is disjoint from supp(𝑚 0), so EVC𝑙𝑘 [𝐾 t 𝐿]
does not depend on any variables that 𝑚 0 depends on. It follows that 𝑚 0 · EVC𝑙𝑘 [𝐾 t 𝐿] is
multilinear and homogeneous of the same degree as 𝑚, and so is 𝑟 in the proof.
Applying Proposition 5.4 repeatedly in a similar way as Proposition 3.6 in the proof of
Lemma 3.7 yields the following formalization of the first step in the setting of Theorem 1.8.
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Lemma 5.5. Let 𝑘, 𝑙, 𝑛, 𝑑 ∈ ℕ , let 𝐶 be a (𝑘 + 𝑑 − 𝑙)-subset of [𝑛], and let 𝐼 denote the ideal generated
by the polynomials EVC𝑙𝑘 [𝐾 t 𝐿] where 𝐾 ranges over all (𝑘 + 1)-subsets of 𝐶 and 𝐿 ranges over all
(𝑙 + 1)-subsets of [𝑛] \ 𝐾. Modulo 𝐼, every multilinear polynomial 𝑝 of degree at most 𝑑 in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ]
is equal to a (𝑘 + 𝑑 − 𝑙, 𝑙)-cored multilinear polynomial with core 𝐶 that is either zero or else has the same
degree as 𝑝.
The following refinement of Lemma 3.8 from the context of Section 3 represents the
corresponding second step in the context of Theorem 1.8. This is where the degree constraint
comes into play.
Lemma 5.6. Let 𝑘, 𝑙, 𝑛, 𝑑 ∈ ℕ with 𝑙 ≤ 𝑑 ≤ 𝑛 − 𝑘. Let 𝑟 be a nonzero multilinear polynomial of degree
𝑑 in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] that is (𝑑 + 𝑘 − 𝑙, 𝑙)-cored. There are disjoint sets 𝐾, 𝐿 ⊆ [𝑛] with |𝐾| = 𝑘 and |𝐿| = 𝑙
so that 𝜕𝐿 𝑟| 𝐾←0 is nonzero upon the substitution (5.1).
Proof. Let 𝐶 denote the core of size at most 𝑑 + 𝑘 − 𝑙, and let 𝑚 be a monomial of 𝑟 of degree 𝑑
that maximizes |supp(𝑚) \ 𝐶|. Let 𝐾 be a subset of [𝑛] \ supp(𝑚) of size 𝑘 that contains all of
𝐶 \ supp(𝑚). Such a set 𝐾 exists because |𝐶 \ supp(𝑚)| = |𝐶| − |supp(𝑚) ∩ 𝐶| ≤ |𝐶| − (𝑑 − 𝑙) ≤ 𝑘,
and |[𝑛] \ supp(𝑚)| = 𝑛 − 𝑑 ≥ 𝑘. Let 𝐿 be a subset of supp(𝑚) of size 𝑙 that contains all of
supp(𝑚) \ 𝐶. Such a set 𝐿 exists because |supp(𝑚) \ 𝐶| ≤ 𝑙 and |supp(𝑚)| = 𝑑 ≥ 𝑙. Note that 𝐾
and 𝐿 are disjoint.
The monomial 𝑚 has a nonzero contribution to 𝜕𝐿 𝑟 | 𝐾←0 . In general, a monomial 𝑚 0 has
a nonzero contribution to 𝜕𝐿 𝑟| 𝐾←0 if and only if supp(𝑚 0) is disjoint from 𝐾 and contains 𝐿.
The disjointness requirement implies that supp(𝑚 0) ∩ 𝐶 ⊆ 𝐶 \ 𝐾 = supp(𝑚) ∩ 𝐶, where the
equality follows from the choice of 𝐾. The inclusion requirement implies that supp(𝑚) \ 𝐶 =
𝐿 \ 𝐶 ⊆ supp(𝑚 0) \ 𝐶, where the equality follows from the choice of 𝐿. In combination with the
maximality of |supp(𝑚) \ 𝐶| among the monomials of 𝑟 of degree 𝑑, this means that either 𝑚 0
does not have degree 𝑑 or else supp(𝑚 0) \ 𝐶 = supp(𝑚) \ 𝐶. It follows that the only monomials
𝑚 0 of 𝑟 of degree 𝑑 that contribute to 𝜕𝐿 𝑟 | 𝐾←0 satisfy supp(𝑚 0) ⊆ supp(𝑚). As 𝑟 only contains
multilinear monomials, 𝑟 has exactly one monomial of degree 𝑑 that has a nonzero contribution
to 𝜕𝐿 𝑟 | 𝐾←0 , namely the monomial 𝑚. We conclude that the polynomial 𝑞(𝑧) that results from
substituting (5.1) into 𝜕𝐿 𝑟 | 𝐾←0 has a nonzero term of degree 𝑑 − 𝑙.
We now have all ingredients to establish Theorem 1.8.
Proof of Theorem 1.8. The necessity of condition 1 and condition 2 for the membership of a
multilinear polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] in Van[RFE𝑙𝑘 ] immediately follows from Proposition 5.1
and Lemma 5.3, respectively. For sufficiency, we need to show that if 𝑝 only contains monomials
of degrees between 𝑙 + 1 and 𝑛 − 𝑘 − 1 and is hit by RFE𝑙𝑘 , then there exist disjoint sets 𝐾, 𝐿 ⊆ [𝑛]
with |𝐾| = 𝑘 and |𝐿| = 𝑙 so that 𝜕𝐿 𝑝| 𝐾←0 is nonzero upon the substitution (5.1).
By Lemma 5.5, we can write 𝑝 as 𝑝 = 𝑞 + 𝑟, where 𝑞 ∈ Van[RFE𝑙𝑘 ] and 𝑟 is a multilinear
(𝑘 + 𝑑 − 𝑙, 𝑙)-cored polynomial that is either zero or else has the same degree as 𝑝. Since
𝑝 ∉ Van[RFE𝑙𝑘 ], the case of zero 𝑟 is ruled out. Thus 𝑟 is a multilinear (𝑘 + 𝑑 − 𝑙, 𝑙)-cored
polynomial of degree 𝑑, where 𝑙 + 1 ≤ 𝑑 ≤ 𝑛 − 𝑘 − 1. Lemma 5.6 then yields disjoint sets
𝐾, 𝐿 ⊆ [𝑛] with |𝐾| = 𝑘 and |𝐿| = 𝑙 so that 𝜕𝐿 𝑟| 𝐾←0 is nonzero upon the substitution (5.1). As
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both 𝑝 and 𝑟 are multilinear, so is 𝑞 = 𝑝 − 𝑟. The contrapositive of Lemma 5.3 implies that
𝜕𝐿 𝑞| 𝐾←0 is zero upon the substitution (5.1). It follows that 𝜕𝐿 𝑝| 𝐾←0 = 𝜕𝐿 𝑞| 𝐾←0 + 𝜕𝐿 𝑟 | 𝐾←0 is
nonzero upon the substitution (5.1).
We conclude this section by detailing the connection between Theorem 1.8 and some prior
applications of the SV generator.
Application to read-once formulas. We start with the theorem that SV1 hits read-once formulas.
The original proof in [41] goes by induction on the depth of 𝐹. The critical part is the inductive
step for the case where the top gate is an addition, say 𝐹 = 𝐹1 + 𝐹2 . The argument in [41] involves
a clever analysis that uses the variable-disjointness of 𝐹1 and 𝐹2 to show that 𝐹1 (SV1 ) and 𝐹2 (SV1 )
cannot cancel each other out. We present an alternate proof that has a similar inductive outline
but follows a more structured, principled approach based on Theorem 1.8 for the critical part.
Theorem 5.7 ([41]). SV1 hits read-once formulas.
Alternate proof. We show by induction on the depth the formula 𝐹 that if 𝐹 is nonconstant, then
so is 𝐹(SV1 ). This suffices because it implies that nonconstant formulas are hit by SV1 , and
nonzero constant formulas are hit as the range of SV1 is nonempty.
The inductive step consists of two cases, depending on whether the top gate is a multiplication
gate or an addition gate. The case of a multiplication gate follows from the general property
that the product of a nonconstant polynomial with any nonzero polynomial is nonconstant. It
remains to consider the case of an addition gate.
For a nonconstant formula 𝐹, 𝐹(SV1 ) is nonconstant iff SV1 hits the variable part of 𝐹 (which
is a nonzero polynomial). By Theorem 1.8 with 𝑘 = 0 and 𝑙 = 1, the latter is the case iff at least
one of the following two conditions hold:
1. 𝐹 has a homogeneous component of degree 1 or at least 𝑛.
2. For some 𝐿 = {𝑖} ⊆ [𝑛], the derivative 𝜕𝑥 𝑖 𝐹 is nonzero upon the substitution (1.7).
Consider a read-once formula 𝐹 with an addition gate on top: 𝐹 = 𝐹1 + 𝐹2 . The variable-
disjointness of 𝐹1 and 𝐹2 implies that if condition 1 holds for at least one of 𝐹1 or 𝐹2 , then it
holds for 𝐹. The same is true for condition 2. The inductive step in the case of an addition gate
at the top follows.
The case of an addition gate in the above proof has a clean geometric interpretation along
the lines of the alternating algebra representation that we discussed in Section 1 for polynomials
that are multilinear (which polynomials computed by read-once formulas are). Recall that we
can think of the variables as vertices, and multilinear monomials as simplices made from those
vertices.2 A multilinear polynomial is a weighted collection of such simplices with weights
from 𝔽 . In this view, Theorem 1.8 translates to the following characterization: a weighted
collection of simplices corresponds to a polynomial in the vanishing ideal of RFE01 iff there
2In this setting the orientation of the simplices does not matter.
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
are no simplices of zero, one, or all vertices (condition 1), and the remaining weights satisfy
a certain system of linear equations (condition 2). Crucially, for each equation in the system,
there is a vertex such that the equation only involves weights of the simplices that contain that
vertex, namely the vertex corresponding to the variable 𝑥 𝑖 where 𝐿 = {𝑖}. Meanwhile, the sum
of two variable-disjoint polynomials corresponds to taking the vertex-disjoint union of two
weighted collections of simplices. It follows directly that if either of the two polynomials violates
a requirement besides the “no simplex of zero vertices” requirement, then their sum violates the
same requirement. The “no simplex of zero vertices” requirement holds automatically when
considering the variable parts, and maps to the special handling of the constant term in the
formal proof.
Zero-substitutions and partial derivatives. As mentioned in the overview, several prior
papers demonstrated the utility of partial derivatives and zero substitutions in the context of
derandomizing PIT using the SV generator, especially for syntactically multilinear models. By
judiciously choosing variables for those operations, these papers managed to simplify 𝑝 and
reduce PIT for 𝑝 to PIT for simpler instances, resulting in an efficient recursive algorithm. Such
recursive arguments can be wrapped into a general framework, similar to the one presented in
[39] for generic 𝑙-independent generators. Whereas the power of the framework in the generic
setting remains open, thanks to Theorem 1.8, we can prove that our framework captures the
full power of the specific 𝑙-independent generator SV𝑙 . More generally, we exhibit a natural
reformulation within the framework of any argument that RFE hits a certain class of multilinear
polynomials, such as those computable with some bounded complexity in some syntactic model.
For the argument, we assume that we can break up the class in the following way.
Ð
Definition 5.8 (grading hypothesis). A class 𝒞 = 𝑘,𝑙∈ℕ 𝒞𝑘,𝑙 of polynomials satisfies the grading
hypothesis if for every 𝑘, 𝑙 ∈ ℕ and 𝑝 ∈ 𝒞𝑘,𝑙 , at least one of the following holds:
• 𝑘 = 𝑙 = 0 and 𝑝 is nonzero.
• 𝑘 > 0 and there is a zero substitution such that the result is in 𝒞𝑘−1,𝑙 .
• 𝑙 > 0 and there is a partial derivative such that the result is in 𝒞𝑘,𝑙−1 .
Under the additional mild assumption of closure under variable rescaling, we obtain a
parameter-efficient framework through direct applications of Theorem 1.8.
Ð
Proposition 5.9. Let 𝒞 = 𝑘,𝑙∈ℕ 𝒞𝑘,𝑙 be a class of polynomials that satisfies the grading hypothesis and
such that each 𝒞𝑘,𝑙 is closed under variable rescaling. If RFE00 hits 𝒞0,0 then RFE𝑙𝑘 hits 𝒞𝑘,𝑙 for every
𝑘, 𝑙 ∈ ℕ .
Proof. The proof is by induction on 𝑘 and 𝑙. The base case is 𝑘 = 𝑙 = 0, where the claim is
immediate. When 𝑘 > 0 or 𝑙 > 0, our hypotheses are such that 𝑝 ∈ 𝒞𝑘,𝑙 either simplifies under a
zero substitution or a partial derivative. In either case, we show how a violation of the conditions
in Theorem 1.8 for a simpler polynomial 𝑝 0 ∈ 𝔽 [𝑥 10 , . . . , 𝑥 0𝑛 ] translates into a corresponding
violation of the conditions for 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], where each variable 𝑥 0𝑖 is a rescaling of 𝑥 𝑖 .
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More specifically, by condition 1 of Theorem 1.8, we may assume that 𝑝 only has homogeneous
components with degrees in the range 𝑙 + 1, . . . , 𝑛 − 𝑘 − 1. We argue in both cases that 𝑝 0 similarly
satisfies condition 1 of Theorem 1.8. By the induction hypothesis and closure under variable
rescaling, it follows that 𝜕0𝐿0 𝑝 0 𝐾0←0 (where the prime in 𝜕0 indicates that the partial derivatives
are with respect to the primed variables 𝑥 0𝑖 ) is nonzero for some 𝐾 0 and 𝐿0 under a particular
substitution. Out of 𝐾 0 and 𝐿0 we then construct 𝐾 and 𝐿 such that 𝜕𝐿 𝑝| 𝐾←0 is nonzero upon the
substitution in condition 2 of Theorem 1.8, where variable rescaling between 𝑥 0𝑖 and 𝑥 𝑖 enables
us to match the substitutions for 𝜕0𝐿0 𝑝 0 𝐾0←0 and 𝜕𝐿 𝑝| 𝐾←0 . We provide the remaining details for
each case separately.
• If 𝑝 simplifies under a zero substitution 𝑥 𝑗 ∗ ← 0, then write 𝑝 as 𝑝 = 𝑞𝑥 𝑗 ∗ + 𝑟 where 𝑞 and
𝑟 are polynomials that do not depend on 𝑥 𝑗 ∗ , and set 𝑝 0(. . . , 𝑥 0𝑖 , . . . ) = 𝑟(. . . , 𝑥 𝑖 , . . . ) with
𝑥 𝑖 = 𝑥 0𝑖 · (𝑎 𝑖 − 𝑎 𝑗 ∗ ). By closure under rescaling, 𝑝 0 ∈ 𝒞𝑘−1,𝑙 , so by induction 𝑝 0 is hit by RFE𝑙𝑘−1 .
We apply Theorem 1.8 to 𝑝 0 with respect to the set of variables {𝑥 10 , . . . , 𝑥 0𝑗 ∗ −1 , 𝑥 0𝑗 ∗ +1 , . . . , 𝑥 0𝑛 }
and 𝑘 replaced by 𝑘 − 1. As 𝑝 only has homogeneous components with degrees in the
range 𝑙 + 1, . . . , 𝑛 − 𝑘 − 1, so does 𝑝 0, and condition 1 of Theorem 1.8 holds for 𝑝 0. This
means that condition 2 does not hold for 𝑝 0. Thus, there must be disjoint 𝐾 0 , 𝐿0 ⊆ [𝑛] \ {𝑗 ∗ }
with |𝐾 0 | = 𝑘 − 1 and |𝐿0 | = 𝑙 so that 𝜕0𝐿0 𝑝 0 𝐾0←0 is nonzero upon the substitution
𝑗∈𝐾 0 (𝑎 𝑖 − 𝑎 𝑗 )
Î
𝑥 0𝑖 ← 𝑧 · Î . (5.2)
𝑗∈𝐿0 (𝑎 𝑖 − 𝑎 𝑗 )
Setting 𝐾 = 𝐾 0 ∪ {𝑗 ∗ } and 𝐿 = 𝐿0, we have
.Ö
𝜕𝐿 𝑝| 𝐾←0 = 𝜕𝐿0 𝑟 | 𝐾0←0 = 𝜕0𝐿0 𝑝 0 𝐾0←0 (𝑎 𝑖 − 𝑎 𝑗 ∗ )
𝑖∈𝐿0
and the substitution (5.2) induces the substitution (1.7).
• If 𝑝 simplifies under a partial derivative 𝜕𝑥 𝑗∗ , then write 𝑝 as 𝑝 = 𝑞𝑥 𝑗 ∗ + 𝑟 where 𝑞 and 𝑟
are polynomials that do not depend on 𝑥 𝑗 ∗ , and set 𝑝 0(. . . , 𝑥 0𝑖 , . . . ) 𝑞(. . . , 𝑥 𝑖 , . . . ) with
𝑘
𝑥 𝑖 = 𝑥 0𝑖 /(𝑎 𝑖 − 𝑎 𝑗 ∗ ). By closure under rescaling, 𝑝 0 ∈ 𝒞𝑘,𝑙−1 , so by induction 𝑝 0 is hit by RFE𝑙−1 .
We apply Theorem 1.8 to 𝑝 0 with respect to the set of variables {𝑥 10 , . . . , 𝑥 0𝑗 ∗ −1 , 𝑥 0𝑗 ∗ +1 , . . . , 𝑥 0𝑛 }
and 𝑙 replaced by 𝑙 − 1. As 𝑝 0 has homogeneous components of degrees one less than 𝑝
does, condition 1 of Theorem 1.8 holds for 𝑝 0, so condition 2 must fail. Thus, there are
disjoint 𝐾 0 , 𝐿0 ⊆ [𝑛] \ {𝑗 ∗ } with |𝐾 0 | = 𝑘 and |𝐿0 | = 𝑙 − 1 so that 𝜕0𝐿0 𝑝 0 𝐾0←0 is nonzero upon
the substitution (5.2). Setting 𝐾 = 𝐾 0 and 𝐿 = 𝐿0 ∪ {𝑗 ∗ }, we have
Ö
𝜕𝐿 𝑝| 𝐾←0 = 𝜕𝐿0 𝑞| 𝐾0←0 = 𝜕0𝐿0 𝑝 0 𝐾0←0 · (𝑎 𝑖 − 𝑎 𝑗 ∗ )
𝑖∈𝐿0
and the substitution (5.2) induces the substitution (1.7).
In both cases we conclude that 𝜕𝐿 𝑝| 𝐾←0 is nonzero upon the substitution (1.7), which is the
sought violation of condition 2 of Theorem 1.8.
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We remark that the mild requirement of closure under variable rescaling in Proposition 5.9
can be dropped completely at the cost of reduced efficiency in parameters.3
Ð
Proposition 5.10. Let 𝒞 = 𝑘,𝑙∈ℕ 𝒞𝑘,𝑙 be a class of polynomials that satisfies the grading hypothesis. If
RFE00 hits 𝒞0,0 then RFE 𝑘+𝑙
𝑘+𝑙 hits 𝒞𝑘,𝑙 for every 𝑘, 𝑙 ∈ ℕ .
Proof sketch. The strategy is the same as in the proof of Proposition 5.9, but in the inductive step
the index 𝑖 ∗ is added to both 𝐾 0 and 𝐿0 instead of just one of the two sets. This obviates the need
for rescaling to ensure that the substitutions match. Note that the resulting sets 𝐾 and 𝐿 are no
longer disjoint, but the general Zoom Lemma accommodates overlapping sets 𝐾 and 𝐿.
Theorem 1.8 tells us that derivatives and zero substitutions suffice to witness when a
multilinear polynomial 𝑝 is hit by SV or RFE. One can ask, if we know more information about
𝑝, can we infer which derivatives and zero substitutions form a witness? In some cases we know.
For example, if 𝑝 has a low-support monomial 𝑥1 · · · 𝑥 𝑙 , then it suffices to take derivatives with
respect to each of 𝑥 1 , . . . , 𝑥 𝑙 . On the other hand, consider that whenever two polynomials 𝑝 and
𝑞 are hit by SV, then so is their product 𝑝𝑞. Given explicit witnesses for 𝑝 and 𝑞, we do not know
how to obtain an explicit witness for the product 𝑝𝑞.
6 Sparseness
By Proposition 3.4, the generators EVC𝑙𝑘 contain exactly 𝑘+𝑙+2
𝑙+1 monomials. The following
result shows that no nonzero polynomial in the vanishing ideal of RFE𝑙𝑘 has fewer monomials.
Corollary 1.6 follows.
Lemma 6.1. Suppose 𝑝 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] is nonzero and has only 𝑠 monomials with nonzero coefficients.
Then, for any 𝑘, 𝑙 such that 𝑘+𝑙+2 > 𝑠, RFE𝑙𝑘 hits 𝑝.
𝑙+1
The tactic here is to show that, if 𝑝 has too few monomials appearing in it, then there is a way to
instantiate the Zoom Lemma wherein 𝑝 𝑑∗ is a single monomial and therefore is nonzero upon
the substitution (4.1).
Proof. For 𝑖 ∈ [𝑛], we define two operations, ↓𝑖 and ↑𝑖 , on nonempty sets of monomials. Applying
↓𝑖 to such a set 𝑀 yields the subset of 𝑀 consisting of the monomials in which 𝑥 𝑖 appears with
its least degree among all the monomials in 𝑀. We define ↑𝑖 similarly, except we select the
monomials in which 𝑥 𝑖 appears with its highest degree. We make the following claim:
Claim 6.2. For any nonempty set of monomials with fewer than 𝑘+𝑙+2
𝑙+1 monomials, there is a sequence of
↓ and ↑ operations, with at most 𝑘 ↓ operations and at most 𝑙 ↑ operations, such that the resulting set of
monomials has exactly one element.
3This is a setting where we exploit the possibility of the sets 𝐾 and 𝐿 in the Zoom Lemma to overlap.
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The claim implies the lemma as follows. Let 𝑀 be the set of monomials with nonzero
coefficient in 𝑝. Apply the claim to 𝑀 to get a sequence of ↓ and ↑ operations resulting in a
single monomial 𝑚0 . Let 𝐾 denote the indices used for the ↓ operations and 𝐿 the indices used
for the ↑ operations. Let 𝑑 ∗ be the degree pattern with domain 𝐾 ∪ 𝐿 that matches 𝑚0 . By how
the operators are defined, every monomial 𝑚 in 𝑀 satisfies either
• deg𝑥 𝑖 (𝑚) > 𝑑 ∗𝑖 for some 𝑖 ∈ 𝐾 (𝑚 was removed by ↓𝑖 ),
• deg𝑥 𝑖 (𝑚) < 𝑑 ∗𝑖 for some 𝑖 ∈ 𝐿 (𝑚 was removed by ↑𝑖 ), or
• deg𝑥 𝑖 (𝑚) = 𝑑 ∗𝑖 for every 𝑖 ∈ 𝐾 ∪ 𝐿, in which case 𝑚 = 𝑚0 .
Accordingly, 𝑑 ∗ is (𝐾, 𝐿)-extremal in 𝑝 and the Zoom Lemma applies. As 𝑝 𝑑∗ is a single monomial,
it is nonzero upon the substitution (4.1). As |𝐾| ≤ 𝑘 and |𝐿| ≤ 𝑙, we conclude that 𝑝 is hit by
RFE𝑙𝑘 .
It remains to prove Claim 6.2. We do this by induction on |𝑀|. In the base case, |𝑀| = 1, in
which case the empty sequence suffices. Otherwise, |𝑀| > 1, in which case there is a variable
𝑥 𝑖 that appears with at least two distinct degrees among monomials in 𝑀. The sets ↓𝑖 (𝑀) and
↑𝑖 (𝑀) are nonempty and disjoint. Since 𝑀 has size less than 𝑘+𝑙+2 = 𝑘+𝑙+1 + 𝑘+𝑙+1
𝑙+1 𝑙+1 𝑙 , either
𝑘+𝑙+1 𝑘+𝑙+1
↓𝑖 (𝑀) has size less than 𝑙+1 , or ↑𝑖 (𝑀) has size less than 𝑙 . Whichever is the case, the
claim follows by applying the inductive hypothesis to it.
7 Set-Multilinearity
Although the generators EVC𝑙𝑘 provided by Theorem 1.3 are not set-multilinear, the vanishing
ideal of RFE𝑙𝑘 does contain set-multilinear polynomials. In this section, we construct some
of degree 𝑙 + 1 with partition classes of size 𝑘 + 2. In fact, we argue that all set-multilinear
polynomials in Van[RFE𝑙𝑘 ] of degree 𝑙 + 1 are in the linear span of the ones we construct.
Our construction is a modification of the one for EVC𝑙𝑘 .
Definition 7.1. Let 𝑘, 𝑙, 𝑛 ∈ ℕ , and let 𝑆1 , . . . , 𝑆 𝑙+1 ⊆ [𝑛] be 𝑙 + 1 disjoint subsets of 𝑘 + 2 indices
each. The polynomial ESMVC𝑙𝑘 is an (𝑙 + 1) × (𝑙 + 1) determinant where each entry is itself a
(𝑘 + 2) × (𝑘 + 2) determinant. We index the rows in the outer determinant by 𝑖 = 1, . . . , 𝑙 + 1,
and the columns by 𝑑 = 𝑙, . . . , 0. In each (𝑖, 𝑑)-th inner matrix, there is one row per 𝑗 ∈ 𝑆 𝑖 ; it is
h i
𝑎 𝑗𝑘 𝑎 𝑗𝑘−1 · · · 𝑎 1𝑗 𝑎 0𝑗 𝑎 𝑑𝑗 𝑥 𝑗 .
The name “ESMVC” is a shorthand for “Elementary Set-Multilinear Vandermonde Circulation”.
Similar to EVC, the precise instantiation of ESMVC requires one to pick an order for the sets
𝑆1 , . . . , 𝑆 𝑙+1 and an order within each set.
Example 7.2. When 𝑘 = 1 and 𝑙 = 2, ESMVC uses three sets of three variables each. To
help convey the structure of the determinant, we name the variable-sets 𝑆1 = {𝑥 1 , 𝑥2 , 𝑥3 },
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𝑆2 = {𝑦1 , 𝑦2 , 𝑦3 }, and 𝑆3 = {𝑧 1 , 𝑧2 , 𝑧3 }, and denote the abscissa of 𝑥 𝑖 by 𝑎 𝑖 , the abscissa of 𝑦 𝑖 by
𝑏 𝑖 , and the abscissa of 𝑧 𝑖 by 𝑐 𝑖 . With this notation and using the index ordering, ESMVC is the
following:
𝑎 11 𝑎 10 𝑎 12 𝑥 1 𝑎 11 𝑎 10 𝑎 11 𝑥 1 𝑎 11 𝑎 10 𝑎 10 𝑥 1
𝑎 21 𝑎 20 𝑎 22 𝑥 2 𝑎 21 𝑎 20 𝑎 21 𝑥 2 𝑎 21 𝑎 20 𝑎 20 𝑥 2
𝑎 31 𝑎 30 𝑎 32 𝑥 3 𝑎 31 𝑎 30 𝑎 31 𝑥 3 𝑎 31 𝑎 30 𝑎 30 𝑥 3
𝑏11 𝑏10 𝑏12 𝑦1 𝑏11 𝑏10 𝑏 11 𝑦1 𝑏11 𝑏10 𝑏10 𝑦1
𝑏21 𝑏 20 𝑏22 𝑦2 𝑏21 𝑏20 𝑏 21 𝑦2 𝑏21 𝑏20 𝑏20 𝑦2 .
𝑏31 𝑏 30 𝑏32 𝑦3 𝑏31 𝑏30 𝑏 31 𝑦3 𝑏31 𝑏30 𝑏30 𝑦3
𝑐11 𝑐 10 𝑐12 𝑧 1 𝑐11 𝑐 10 𝑐11 𝑧 1 𝑐 11 𝑐10 𝑐10 𝑧 1
𝑐21 𝑐20 𝑐22 𝑧 2 𝑐21 𝑐 20 𝑐21 𝑧 2 𝑐 21 𝑐20 𝑐20 𝑧 2
𝑐31 𝑐30 𝑐32 𝑧 3 𝑐31 𝑐 30 𝑐31 𝑧 3 𝑐 31 𝑐30 𝑐30 𝑧 3
The elementary properties of EVC𝑙𝑘 from Proposition 3.4 extend as follows to ESMVC𝑙𝑘 .
Proposition 7.3. For any 𝑘, 𝑙 ∈ ℕ and index sets 𝑆1 , . . . , 𝑆 𝑙+1 as in Definition 7.1, ESMVC𝑙𝑘 is
skew-symmetric with respect to the order of the sets 𝑆1 , . . . , 𝑆 𝑙+1 , and the choice of order within each
set, in that any permutation thereof changes the construction by merely multiplying by the sign of the
permutation. For any order, ESMVC𝑙𝑘 is nonzero, homogeneous of degree 𝑙 + 1, and set-multilinear with
respect to the partition 𝑆1 , . . . , 𝑆 𝑙+1 , and every monomial consistent with the partitions appears with a
nonzero coefficient. When the sets are ordered as 𝑆1 , . . . , 𝑆 𝑙+1 and the variables associated with 𝑆 𝑖 are
labeled and ordered as (𝑥 𝑖,1 , . . . , 𝑥 𝑖,𝑘+2 ) for 𝑖 = 1, . . . , 𝑙 + 1, the coefficient of 𝑥 1,1 · · · · · 𝑥 𝑙+1,1 equals
𝑙 𝑘
𝑎 1,1 ··· 𝑎 1,1 𝑙+1 𝑎 𝑖,2 ··· 𝑎 0𝑖,2
0
. .. .. . .. .. .
Ö
(−1) (𝑘+1)(𝑙+1)
· .. . . · .. . . (7.1)
𝑙 𝑖=1 𝑘
𝑎 𝑙+1,1 · · · 𝑎 𝑙+1,1
0
𝑎 𝑖,𝑘+2 · · · 𝑎 0𝑖,𝑘+2
Proof. All assertions to be proved follow from elementary properties of determinants, that
Vandermonde determinants are nonzero unless they have duplicate rows, and the following
computation for the coefficient of 𝑥 1,1 · · · · · 𝑥 𝑙+1,1 : Plug 1 into 𝑥 𝑖,1 for 𝑖 = 1, . . . , 𝑙 + 1 and 0 into the
remaining variables, and minor expand along the last column each of the inner determinants.
Due to the minor expansions, the elements in the 𝑖-th row of the outer determinant have a
common factor of (−1) 𝑘+1 times the (𝑘 + 1) × (𝑘 + 1) determinant for that value of 𝑖 in the product
on the right-hand side of (7.1). After removing those common factors from all 𝑙 + 1 rows, the
remaining (𝑙 + 1) × (𝑙 + 1) outer determinant equals the determinant in the middle of (7.1).
The following theorem formalizes the role ESMVC plays among the degree-(𝑙+1) polynomials
with respect to Van[RFE𝑙𝑘 ].
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Theorem 7.4. Let 𝑘, 𝑙 ∈ ℕ and let 𝑋1 , . . . , 𝑋𝑙+1 be 𝑙 + 1 disjoint sets of indices (of any size). The linear
span of ESMVC𝑙𝑘 [𝑆1 , . . . , 𝑆 𝑙+1 ], over all choices of 𝑆 𝑖 ⊆ 𝑋𝑖 with |𝑆 𝑖 | = 𝑘 + 2, equals the set-multilinear
polynomials in Van[RFE𝑙𝑘 ] with variable partition (𝑋1 , . . . , 𝑋𝑙+1 ).
Theorem 7.4 and Proposition 7.3 imply Corollary 1.7 that there are no set-multilinear
polynomials of degree 𝑙 + 1 in Van[RFE𝑙𝑘 ] that have at least one partition 𝑋𝑖 of size less than 𝑘 + 2.
The proof of Theorem 7.4 follows the same outline as the one of Theorem 1.3 in Section 3. We
start by showing that all instantiations of ESMVC𝑙𝑘 are contained in Van[RFE𝑙𝑘 ] using a similar
argument as that for EVC𝑙𝑘 .
Lemma 7.5. For every 𝑘, 𝑙 ∈ ℕ and every choice of 𝑙 + 1 disjoint sets 𝑆1 , . . . , 𝑆 𝑙+1 of 𝑘 + 2 indices each,
ESMVC𝑙𝑘 [𝑆1 , . . . , 𝑆 𝑙+1 ] vanishes at RFE𝑙𝑘 .
Proof. Let 𝑔/ℎ be a seed for RFE𝑙𝑘 . Let 𝐴 be the (𝑙 + 1) × (𝑙 + 1) outer matrix defining ESMVC, so
that ESMVC det(𝐴). Recall that the columns of 𝐴 are indexed by 𝑑 = 𝑙, . . . , 0. Let ℎ® ∈ 𝔽 𝑙+1
be the column vector where the row indexed by 𝑑 is the coefficient of 𝛼 𝑑 in ℎ(𝛼). We show
that, after substituting RFE𝑙𝑘 (𝑔/ℎ), the matrix-vector product 𝐴 ℎ® ∈ 𝔽 𝑙+1 yields the zero vector.
It follows that evaluating ESMVC at RFE𝑙𝑘 (𝑔/ℎ) vanishes, as it is the determinant of a singular
matrix.
Fix 𝑖 ∈ {1, . . . , 𝑙 + 1}, and focus on the 𝑖-th coordinate of 𝐴 ℎ. ® The (𝑖, 𝑑) entry of 𝐴 is a
determinant; let 𝐵 𝑖,𝑑 be the inner matrix as in Definition 7.1. As 𝑑 varies, only the last column
of 𝐵 𝑖,𝑑 changes. Thus, by multilinearity of the determinant, the 𝑖-th entry of 𝐴 ℎ® is itself a
determinant. Recalling that the rows of 𝐵 𝑖,𝑙 , . . . , 𝐵 𝑖,0 are indexed by 𝑗 ∈ 𝑋𝑖 , the 𝑗-th row of this
determinant is h i
𝑎 𝑗𝑘 · · · 𝑎 0𝑗 ℎ(𝑎 𝑗 )𝑥 𝑗 .
After substituting RFE𝑙𝑘 (𝑔/ℎ), it becomes
h i
𝑎 𝑗𝑘 ··· 𝑎 0𝑗 𝑔(𝑎 𝑗 ) .
Since 𝑔 is a degree-𝑘 polynomial, the columns of 𝐵 𝑖,𝑑 are linearly dependent, so the determinant
is zero.
Next, we argue that every polynomial in Van[RFE𝑙𝑘 ] that is set-multilinear with respect to
the variable partition (𝑋1 , . . . , 𝑋𝑙+1 ) is in the ideal 𝐼 generated by the instantiations of ESMVC𝑙𝑘
in the statement of Theorem 7.4. We use a similar two-step approach as for Theorem 1.3 in
Section 3.
1. Modulo the ideal 𝐼, every polynomial 𝑝 is equal to a polynomial 𝑟 (depending on 𝑝) with
a certain structure (Lemma 7.7).
2. Every nonzero polynomial 𝑟 that has the structure and is is set-multilinear with respect to
the variable partition (𝑋1 , . . . , 𝑋𝑙+1 ) is hit by RFE𝑙𝑘 (Lemma 7.8).
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For step 1, we need a suitable replacement for being (𝑐, 𝑡)-cored. The following adaptation
to the set-multilinear setting suffices.
Definition 7.6. Let 𝑋1 , . . . , 𝑋𝑑 ⊆ [𝑛] be disjoint sets of indices. A polynomial that is set-
multilinear with respect to the partition (𝑋1 , . . . , 𝑋𝑑 ) is (𝑐, 𝑡)-multi-cored if there exists a set
𝐶 𝐶1 t · · · t 𝐶 𝑑 , with 𝐶 𝑖 ⊆ 𝑋𝑖 , |𝐶 𝑖 | ≤ 𝑐, such that every monomial 𝑚 of the polynomial satisfies
|supp(𝑀) \ 𝐶| ≤ 𝑡.
We refer to the set 𝐶 in Definition 7.6 as a multi-core.
Lemma 7.7. Let 𝑘, 𝑙 ∈ ℕ and let 𝑋1 , . . . , 𝑋𝑙+1 ⊆ [𝑛] be disjoint sets of indices. Suppose 𝐶
𝐶1 t · · · t 𝐶 𝑙+1 is a set of indices such that 𝐶 𝑖 ⊆ 𝑋𝑖 and |𝐶 𝑖 | = 𝑘 + 1. Let 𝐼 be the ideal generated by the
polynomials ESMVC𝑙𝑘 [𝐶1 t {𝑗1 }, . . . , 𝐶 𝑙+1 t {𝑗𝑙+1 }], where 𝑗𝑖 ∈ 𝑋𝑖 \ 𝐶 𝑖 . Modulo 𝐼, every set-multilinear
polynomial with respect to the variable partition 𝑋1 , . . . , 𝑋𝑙+1 equals a (𝑘 + 1, 𝑙)-multi-cored polynomial
with multi-core 𝐶.
Proof. By linearity it suffices to establish the result for any monomial 𝑚 that is set-multilinear
with respect to the partition (𝑋1 , . . . , 𝑋𝑙+1 ). If supp(𝑚)∩𝐶 is nonempty, then 𝑚 is already (𝑘 +1, 𝑙)-
multi-cored with multi-core 𝐶 because 𝑚 only has 𝑙 + 1 variables in its support. Otherwise, let
𝑚 = 𝑥 𝑗1 · · · 𝑥 𝑗𝑙+1 . By Proposition 7.3, the polynomial ESMVC𝑙𝑘 [𝐶1 t {𝑗1 }, . . . , 𝐶 𝑙+1 t {𝑗𝑙+1 }] can
be written as 𝑐 · 𝑚 + 𝑟 where 𝑐 ∈ 𝔽 is nonzero and 𝑟 is a linear combination of monomials 𝑚 0
that are set-multilinear with respect to the partition (𝑋1 , . . . , 𝑋𝑙+1 ) and such that supp(𝑚 0) ∩ 𝐶
is nonempty. The result for 𝑚 follows by writing 𝑚 ≡ −𝑐 −1 · 𝑟 mod 𝐼.
Step 2 is another application of the Zoom Lemma. We make use of the version geared
towards multilinear polynomials, namely Lemma 5.3.
Lemma 7.8. Let 𝑘, 𝑙 ∈ ℕ and let 𝑋1 , . . . , 𝑋𝑙+1 ⊆ [𝑛] be disjoint sets of indices. Every nonzero polynomial
that is set-multilinear with respect to the partition (𝑋1 , . . . , 𝑋𝑙+1 ) and that is (𝑘 + 1, 𝑙)-multi-cored is hit
by RFE𝑙𝑘 .
Proof. Let 𝑟 satisfy the hypotheses of the lemma with multi-core 𝐶. Let 𝑚 ∗ be a monomial
in 𝑟 for which supp(𝑚 ∗ ) \ 𝐶 is maximal with respect to inclusion. Such a monomial exists
because 𝑟 is nonzero. Let 𝑗 ∗ ∈ supp(𝑚 ∗ ) ∩ 𝐶. Such an index exists since |supp(𝑚 ∗ )| = 𝑙 + 1
and |supp(𝑚 ∗ ) \ 𝐶 | ≤ 𝑙 by the multi-core property. Let 𝑖 ∗ ∈ [𝑙 + 1] be such that 𝑗 ∗ ∈ 𝑋𝑖 ∗ . Set
𝐾 𝐶 ∩ 𝑋𝑖 ∗ \ {𝑗 ∗ } and 𝐿 supp(𝑚 ∗ ) \ {𝑗 ∗ }. Note that |𝐾| ≤ (𝑘 + 1) − 1 = 𝑘 and |𝐿| ≤ (𝑙 + 1) − 1 = 𝑙.
By set-multilinearity, monomials 𝑚 in 𝑟 for which 𝜕𝐿 𝑚| 𝐾←0 is nonzero need to have the form
𝑥 𝑗 · 𝑥 𝐿 where 𝑗 ∈ 𝑋𝑖 ∗ \ 𝐾. The monomial 𝑚 ∗ is of the form with 𝑗 = 𝑗 ∗ . By the maximality of
𝑚 ∗ , any monomial in 𝑟 of the form has to have 𝑗 ∈ 𝐶. Since 𝐶 ∩ (𝑋𝑖 ∗ \ 𝐾) = {𝑗 ∗ }, it follows that
𝑚 ∗ is the only monomial in 𝑟 that contributes to 𝜕𝐿 𝑟| 𝐾←0 . Since 𝜕𝐿 𝑚 ∗ | 𝐾←0 = 𝑥 𝑗 ∗ , it follows that
𝜕𝐿 𝑟 | 𝐾←0 is nonzero upon the substitution (5.1). We conclude that RFE𝑙𝑘 hits 𝑟 by Lemma 5.3.
We now have all ingredients to establish Theorem 7.4.
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Proof of Theorem 7.4. Let 𝒮 (𝑆1 , . . . , 𝑆 𝑙+1 ) range as in the statement. The linear span of the
polynomials ESMVC𝑙𝑘 [𝒮] is set-multilinear with respect to the variable partition (𝑋1 , . . . , 𝑋𝑙+1 )
by Proposition 7.3, and in Van[RFE𝑙𝑘 ] by Lemma 7.5. In the other direction, the combination of
Lemma 7.7 and Lemma 7.8 imply that every polynomial 𝑝 ∈ Van[RFE𝑙𝑘 ] that is set-multilinear
with respect to the variable partition (𝑋1 , . . . , 𝑋𝑙+1 ) falls inside the ideal 𝐼 generated by the poly-
nomials ESMVC𝑙𝑘 [𝒮], i. e., 𝑝 = 𝒮 𝑞𝒮 ESMVC𝑙𝑘 [𝒮] for some polynomials 𝑞 𝒮 . As all polynomials
Í
ESMVC𝑙𝑘 [𝒮] as well as 𝑝 are homogeneous of degree 𝑙 + 1, it follows that each 𝑞𝒮 can be replaced
by its constant term.
8 Read-Once Oblivious Algebraic Branching Programs
In this section we provide some background on ROABPs and establish Theorem 1.9.
8.1 Background
Algebraic branching programs are a syntactic model for algebraic computation. One forms
a directed graph with a designated source and sink. Each edge is labeled by a polynomial
that depends on at most one variable among 𝑥 1 , . . . , 𝑥 𝑛 . The branching program computes a
polynomial in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] by summing, over all source-to-sink paths, the product of the labels
on the edges of each path.
A special subclass of algebraic branching programs are read-once oblivious algebraic
branching programs (ROABPs). In this model, the vertices of the branching program are
organized in layers. The layers are totally ordered, and edges exist only from one layer to the
next. For each variable, there is at most one consecutive pair of layers between which that
variable appears, and for each pair of consecutive layers, there is at most one variable that
appears between them. In this way, every source-to-sink path reads each variable at most once
(the branching program is read-once), and the order in which the variables are read is common
to all paths (the branching program is oblivious). We can always assume that the number of
layers equals one plus the number of variables under consideration.
The number of vertices comprising a layer is called its width. The width of an ROABP is the
largest width of its layers. The minimum width of an ROABP computing a given polynomial
can be characterized in terms of the rank of coefficient matrices constructed as follows.
Definition 8.1. Let 𝑈 t 𝑉 = [𝑛] be a partition of the variable indices, and let 𝑀𝑈 and 𝑀𝑉 be the
sets of monomials that are supported on variables indexed by 𝑈 and 𝑉, respectively. For any
polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] define the matrix
CMat𝑈 ,𝑉 (𝑝) ∈ 𝔽 𝑀𝑈 ×𝑀𝑉
by setting the (𝑚𝑈 , 𝑚𝑉 ) entry to equal the coefficient of 𝑚𝑈 𝑚𝑉 in 𝑝.
CMat𝑈 ,𝑉 (𝑝) is formally an infinite matrix, but it has only finitely many nonzero entries.
When 𝑝 has degree at most 𝑑, one can just as well truncate CMat𝑈 ,𝑉 (𝑝) to include only rows
and columns indexed by monomials of degree at most 𝑑.
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Lemma 8.2 ([42]). Let 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] be any polynomial. There is an ROABP of width 𝑤 computing
𝑝 in the variable order 𝑥 1 , . . . , 𝑥 𝑛 if and only if, for every 𝑠 ∈ {0, . . . , 𝑛}, with respect to the partition
𝑈 = {1, . . . , 𝑠} and 𝑉 = {𝑠 + 1, . . . , 𝑛}, we have
rank(CMat𝑈 ,𝑉 (𝑝)) ≤ 𝑤.
We group the monomials in 𝑀𝑈 and 𝑀𝑉 by their degrees and order the groups by increasing
degree. This induces a block structure on CMat𝑈 ,𝑉 (𝑝) with one block for every choice of 𝑟, 𝑐 ∈ ℕ ;
the (𝑟, 𝑐) block is the submatrix with rows indexed by degree-𝑟 monomials in 𝑀𝑈 and columns
indexed by degree-𝑐 monomials in 𝑀𝑉 . In the case where 𝑝 is homogeneous, the only nonzero
blocks occur for 𝑟 + 𝑐 equal to the degree of 𝑝. In this case the rank of CMat𝑈 ,𝑉 (𝑝) is the sum of
the ranks of its blocks.
In general, the rank of CMat𝑈 ,𝑉 (𝑝) is at least the rank of CMat𝑈 ,𝑉 (𝑝 (min) ), where 𝑝 (min)
denotes the homogeneous component of 𝑝 of the lowest degree, 𝑑min . This follows because the
submatrix of CMat𝑈 ,𝑉 (𝑝) consisting of the rows and columns indexed by monomials of degree
at most 𝑑min has a block structure that is triangular with the blocks of CMat𝑈 ,𝑉 (𝑝 (min) ) on the
hypotenuse. The observation yields the following folklore consequence of Lemma 8.2.
Proposition 8.3. Let 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] be any nonzero polynomial, and let 𝑝 (min) be the nonzero
homogeneous component of 𝑝 of least degree. If 𝑝 can be computed by an ROABP of width 𝑤, then so can
𝑝 (min) .
8.2 Hitting property / lower bound
We now prove the ROABP hitting property of SV given in Theorem 1.9 and the equivalent
ROABP lower bound given in Theorem 1.10. Both theorems follow from the next statement in a
standard way.
Theorem 8.4. For any integer 𝑙 ≥ 1, every nonzero multilinear homogeneous polynomial of degree 𝑙 + 1
in the vanishing ideal of SV𝑙 requires ROABP width at least (𝑙/3) + 1.
For completeness, before proving Theorem 8.4, we argue how our ROABP hitting property
and lower bound follow.
Proof of Theorem 1.9 and Theorem 1.10. The theorems are equivalent by complementation. We
explain how Theorem 1.9 follows from Theorem 8.4.
Fix 𝑝 satisfying the hypotheses of Theorem 1.9. We show that RFE𝑙−1𝑙 hits 𝑝; this implies SV𝑙
𝑙−1 𝑙
hits 𝑝 because RFE𝑙 and SV are equivalent up to variable rescaling, and rescaling variables
does not affect ROABP width.
If 𝑝 contains a monomial depending on at most 𝑙 variables, then Proposition 5.1 implies
that RFE𝑙−1
𝑙 hits 𝑝. The remaining case is when the homogeneous component 𝑝 (min) of the least
degree is multilinear of degree 𝑙 + 1. By Proposition 8.3, 𝑝 (min) has ROABP width less than
(𝑙/3) + 1. By Theorem 8.4, 𝑝 (min) is hit by RFE𝑙−1
𝑙 , and by Proposition 5.2 so is 𝑝.
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In the remainder of this section we establish Theorem 8.4. We do not try to optimize the
dependence of the bound on 𝑙.
Fix a positive integer 𝑙, and fix an arbitrary variable order, say 𝑥 1 , . . . , 𝑥 𝑛 . We show that, for
every polynomial 𝑝 that is nonzero, multilinear, and homogeneous of degree 𝑙 + 1, and that
belongs to the vanishing ideal of RFE𝑙−1 𝑙 , there exists some 𝑠 ∈ {0, . . . , 𝑛} so that, with respect
to the partition 𝑈 = {1, . . . , 𝑠}, 𝑉 = {𝑠 + 1, . . . , 𝑛}, it holds that rank(CMat𝑈 ,𝑉 (𝑝)) ≥ (𝑙/3) + 1.
Theorem 8.4 then follows by Lemma 8.2.
Let 𝐶 CMat𝑈 ,𝑉 (𝑝). As 𝑝 is homogeneous of degree 𝑙 + 1, 𝐶 is block diagonal, with a block
𝐶 𝑑 for each 𝑑 ∈ {0, . . . , 𝑙 + 1} consisting of the rows indexed by monomials of degree 𝑑 and
the columns indexed by monomials of degree 𝑙 + 1 − 𝑑. The block diagonal structure implies
rank(𝐶) = 𝑙+1
Í
𝑑=0 rank(𝐶 𝑑 ).
Via condition 2 of Theorem 1.8, the hypothesis that 𝑝 belongs to Van[RFE𝑙−1 𝑙 ] induces linear
equations on the entries in the blocks 𝐶 𝑑 . For homogeneous polynomials like 𝑝, the condition
stipulates that for all disjoint subsets 𝐾, 𝐿 ⊆ [𝑛] with |𝐾| = 𝑘 = 𝑙 − 1 and |𝐿| = 𝑙, 𝜕𝐿 𝑝| 𝐾←0
vanishes at the point (1.7) with 𝑧 = 1. This is a linear equation in the coefficients of 𝜕𝐿 𝑝| 𝐾←0 ,
which are entries in the blocks 𝐶 𝑑 of 𝐶. In fact, each of these equations only reads entries from
two adjacent blocks, i. e., blocks 𝐶 𝑑 and 𝐶 𝑑0 with |𝑑 − 𝑑0 | = 1. This is because 𝐿 has size 𝑙, one less
than the degree of 𝑝, so the only monomials that contribute to 𝜕𝐿 𝑝| 𝐾←0 are those that are one
variable 𝑥 𝑖 times the product of the variables indexed by 𝐿. It follows that the corresponding
linear equation on 𝐶 reads only entries that reside in the blocks 𝐶 |𝐿∩𝑈 |+1 (for 𝑖 ∈ 𝑈) and 𝐶 |𝐿∩𝑈 |
(for 𝑖 ∈ 𝑉).
We exploit the structure of these equations and argue that, for an appropriate choice of the
partition index 𝑠, rank(𝐶) is high.
Ingredients. Our analysis has four ingredients. The first ingredient is the fact that rank(𝐶) is
at least the number of nonzero blocks 𝐶 𝑑 . This is because a nonzero block has rank at least 1,
and rank(𝐶) is the sum of the ranks of the blocks. This simple observation means we can focus
on situations where relatively few of the blocks are nonzero.
The second ingredient establishes an alternative lower bound on rank(𝐶) in terms of the
minimum distance between a nonzero block 𝐶 𝑑 and either extreme (𝑑 = 0 or 𝑑 = 𝑙 + 1). Another
way to think about this distance is as the maximum 𝑡 such that every monomial in 𝑝 depends
on at least 𝑡 variables indexed by 𝑈 and at least 𝑡 variables indexed by 𝑉.
Lemma 8.5. Let 𝑝 ∈ Van[RFE𝑙−1 𝑙 ] be nonzero, multilinear, and homogeneous of degree 𝑙 + 1, let 𝑈 t 𝑉
be a partition of [𝑛], and let 𝐶 CMat𝑈 ,𝑉 (𝑝). If every monomial in p depends on at least 𝑡 variables
indexed by 𝑈 and at least 𝑡 variables indexed by 𝑉, then rank(𝐶) ≥ 𝑡 + 1.
The proof involves revisiting the equations from condition 2 of Theorem 1.8 and modifying4
the underlying instantiations of Lemma 4.3 to obtain a system of linear equations with a simple
enough structure that we can analyze.
4This is a setting where we exploit the possibility of the sets 𝐾 and 𝐿 in the Zoom Lemma to overlap.
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The remaining ingredients allow us to reduce to situations where either the first or second
ingredient applies. The third ingredient lets us fix any two zero blocks and zero out all the
blocks that are not between them.
Proposition 8.6. Let 𝑝 ∈ Van[RFE𝑙−1 𝑙 ] be multilinear and homogeneous of degree 𝑙 + 1. Let 𝑈 t 𝑉 be a
partition of [𝑛], and let 𝐶 CMat𝑈 ,𝑉 (𝑝). Suppose that for some 𝑑1 , 𝑑2 ∈ {−1, . . . , 𝑙 + 2} with 𝑑1 ≤ 𝑑2 ,
we have 𝐶 𝑑1 = 0 and 𝐶 𝑑2 = 0, where 𝐶−1 0 and 𝐶 𝑙+2 0. Let 𝑝 0 be the polynomial obtained from 𝑝 by
zeroing out the blocks 𝐶 𝑑 with 𝑑 < 𝑑1 or 𝑑 > 𝑑2 . Then 𝑝 0 belongs to Van[RFE𝑙𝑙−1 ].
As zeroing out blocks does not increase the rank of 𝐶, our lower bound for rank(𝐶) reduces
to the same lower bound for the rank of CMat𝑈 ,𝑉 (𝑝 0). This effectively extends the scope of the
second ingredient: Alone, the second ingredient requires that all nonzero blocks of 𝐶 be far
from the extremes; with the third ingredient, it suffices that there exists a subinterval of nonzero
blocks that is surrounded by zero blocks and that is far from the extremes. The proof hinges on
the adjacent-block property of the equations from condition 2 of Theorem 1.8.
The ingredients thus far suffice provided there exists a nonzero block far from the extremes:
Such a block belongs to some subinterval of nonzero blocks that is surrounded by zero blocks,
say 𝐶 𝑑1 to the left and 𝐶 𝑑2 to the right, and the subinterval either is large and therefore has many
nonzero blocks such that the first ingredient applies, or else it is small and therefore stays far
from the extremes such that the combination of the second and third ingredients applies. See
Figure 1 for an illustration. The fourth and final ingredient lets us ensure there is a nonzero
block far from the extremes by setting the partition index 𝑠 appropriately. In fact, it lets us
guarantee a zero-to-nonzero transition at a position of our choosing.
Proposition 8.7. For every 𝑑 ∈ {−1, . . . , 𝑙}, there is 𝑠 ∈ {0, . . . , 𝑛} such that 𝐶 𝑑 = 0 and 𝐶 𝑑+1 ≠ 0
with respect to the partition 𝑈 = {1, . . . , 𝑠}, 𝑉 = {𝑠 + 1, . . . , 𝑛}, where 𝐶−1 0.
Combining ingredients. Let us find out what lower bound on rank(𝐶) the prior ingredients
give us as a function of the position 𝑑 = 𝑑1 in the interval where we have a guaranteed zero-to-
nonzero transition as in Proposition 8.7. Starting from position 𝑑1 , keep increasing the position
index until we hit the next zero block, say at position 𝑑2 , where we use 𝐶 𝑙+2 0 as a sentinel.
See Figure 1.
1. By the first ingredient, since the middle interval consists of nonzero blocks only, rank(𝐶) ≥
𝑑2 − 𝑑1 − 1.
2. By the combination of the second and the third ingredient, we have that rank(𝐶) ≥ 𝑡 + 1
where 𝑡 = min(𝑑1 + 1, 𝑙 + 2 − 𝑑2 ) is the minimum length of the leftmost and rightmost
intervals. Indeed, let 𝑝 0 be the polynomial obtained from 𝑝 by zeroing out the blocks 𝐶 𝑑
with 𝑑 < 𝑑1 or 𝑑 > 𝑑2 . By Proposition 8.6 𝑝 0 ∈ Van[RFE𝑙−1
𝑙 ]. The polynomial 𝑝 is nonzero
0
as it contains the original block 𝐶 𝑑+1 , which is nonzero. It is homogeneous of degree
𝑙 + 1 and multilinear as all of its monomials also occur in the homogeneous multilinear
polynomial 𝑝 of degree 𝑙 + 1. By construction, every monomial in 𝑝 0 contains at least
𝑑1 + 1 variables indexed by 𝑈, and at least 𝑙 + 2 − 𝑑2 variables indexed by 𝑉. As such,
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
𝐶𝑑 0 ∗ ∗ 0 ≠0 ≠0 ≠0 0 ∗ ∗ 0
𝐶 𝑑0 0 0 0 0 ≠0 ≠0 ≠0 0 0 0 0
𝑑 −1 0 ··· 𝑑1 𝑑1 + 1 ··· 𝑑2 − 1 𝑑2 ··· 𝑙+1 𝑙+2
leftmost interval middle interval rightmost interval
𝑑1 + 1 blocks 𝑑2 − 𝑑1 − 1 blocks 𝑙 + 2 − 𝑑2 blocks
Figure 1: Rank lower bound analysis in terms of the blocks 𝐶 𝑑 of 𝑝 and 𝐶 𝑑0 of 𝑝 0 (Proposition 8.6)
𝑝 0 satisfies the conditions of Lemma 8.5 with 𝑡 = min(𝑑1 + 1, 𝑙 + 2 − 𝑑2 ). It follows that
rank(𝐶) ≥ rank(CMat𝑈 ,𝑉 (𝑝 0)) ≥ 𝑡 + 1.
If the rightmost interval has length at least the leftmost interval (𝑙 + 2 − 𝑑2 ≥ 𝑑1 + 1), then
item 2 yields rank(𝐶) ≥ 𝑑1 + 2. Otherwise, the rightmost interval is strictly shorter than
the leftmost interval (𝑑1 + 1 > 𝑙 + 2 − 𝑑2 ); this implies that the middle interval has length
at least 𝑙 − 2𝑑1 + 1, which by item 1 yields rank(𝐶) ≥ 𝑙 − 2𝑑1 + 1. In any case, the bound
rank(𝐶) ≥ min(𝑑1 + 2, 𝑙 − 2𝑑1 + 1) holds. Taking 𝑑1 = 𝑙−13 optimizes this expression, achieving
rank(𝐶) ≥ 𝑙−1
3 + 2 ≥ (𝑙/3) + 1. This completes the proof of Theorem 8.4 modulo the proofs of
ingredients two through four.
Proofs. We conclude by proving ingredients two through four. We start with the one that
requires the least specificity (ingredient 4, Proposition 8.7), then do ingredient 3 (Proposition 8.6),
and end with the one that involves the most structure (ingredient 2, Lemma 8.5).
Proof of Proposition 8.7. When 𝑠 = 0, 𝐶0 contains all entries. As 𝑠 increases by 1, some entries
move from their current block 𝐶 𝑑0 to the next block 𝐶 𝑑0+1 . Finally, when 𝑠 = 𝑛, 𝐶 𝑙+1 contains all
entries. For 𝑑 ≥ 0, it follows that every nonzero entry moves from 𝐶 𝑑 to 𝐶 𝑑+1 at some time. If
we stop increasing 𝑠 right after the last nonzero entry of 𝐶 moves out of 𝐶 𝑑 , we have 𝐶 𝑑 = 0 and
𝐶 𝑑+1 ≠ 0. For 𝑑 = −1, we can pick 𝑠 = 0 as 𝐶−1 = 0 and 𝐶0 = 𝐶 ≠ 0.
Proof of Proposition 8.6. It suffices to show that whenever 𝑝 satisfies the two conditions in
Theorem 1.8, then so does 𝑝 0. Both 𝑝 and 𝑝 0 are homogeneous. Condition 1 holds for 𝑝 0 as
𝑝 0 either is zero or else has the same degree as 𝑝. Regarding condition 2, as mentioned, the
condition is equivalent to a system of homogeneous linear equations on 𝐶 0 CMat𝑈 ,𝑉 (𝑝 0), each
involving only an adjacent pair of blocks in 𝐶 0. Those that involve only blocks 𝐶 𝑑0 with 𝑑 ≤ 𝑑1
are met as the equations are homogeneous and the involved blocks are all zero. The same holds
for the equations that involve only blocks 𝐶 𝑑0 with 𝑑 ≥ 𝑑2 . The remaining equations involve only
blocks 𝐶 𝑑0 with 𝑑 ∈ {𝑑1 , . . . , 𝑑2 }, on which 𝑝 and 𝑝 0 agree. As the equations hold for 𝐶, they also
hold for 𝐶 0.
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It remains to argue Lemma 8.5. Our proof makes use of linear equations that are closely
related to those given by Theorem 1.8, which in turn come from the Zoom Lemma. We revisit
the application of the Zoom Lemma so as to obtain a simpler coefficient matrix—ultimately
a Cauchy matrix—that enables a deeper analysis. To facilitate the discussion, we utilize the
following notation. As 𝑝 is multilinear, we only need to consider rows indexed by monomials of
the form 𝑖∈𝐼 𝑥 𝑖 for 𝐼 ⊆ 𝑈 and columns indexed by monomials of the form 𝑗∈𝐽 𝑥 𝑗 for 𝐽 ⊆ 𝑉.
Î Î
This allows us to index rows by subsets 𝐼 ⊆ 𝑈 and columns by subsets 𝐽 ⊆ 𝑉. For 𝐼 ⊆ 𝑈 and
𝐽 ⊆ 𝑉 we denote by 𝐶(𝐼, 𝐽) the corresponding entry of 𝐶. The following proposition describes
the linear equations we use.
Proposition 8.8. Let 𝑝 ∈ Van[RFE𝑙−1𝑙 ] be multilinear, and homogeneous of degree 𝑙 + 1, let 𝑈 t 𝑉 be a
partition of [𝑛], and let 𝐶 CMat𝑈 ,𝑉 (𝑝). For every 𝐼 ⊆ 𝑈 and 𝐽 ⊆ 𝑉 with |𝐼 | + |𝐽 | = 𝑙, and for every
𝑗 ∗ ∈ 𝐼 ∪ 𝐽,
Õ 𝐶({𝑖} ∪ 𝐼, 𝐽) Õ 𝐶(𝐼, {𝑖} ∪ 𝐽)
+ = 0. (8.1)
𝑎 𝑖 − 𝑎 𝑗∗ 𝑎 𝑖 − 𝑎 𝑗∗
𝑖∈𝑈\𝐼 𝑖∈𝑉\𝐽
Proof. Set 𝐿 𝐼 ∪ 𝐽 and 𝐾 𝐿 \ {𝑗 ∗ }, and note that 𝐾 ⊆ 𝐿. Let 𝑑 ∗ ∈ ℕ 𝐿 be the all-1 degree pattern
with domain 𝐿, and let 𝑚 ∗ 𝑖∈𝐿 𝑥 𝑖 be the monomial supported on 𝐿 that matches 𝑑 ∗ . As 𝑝 is
Î
multilinear, 𝑑 ∗ is (𝐾, 𝐿)-extremal in 𝑝. Since 𝑝 is in Van[RFE𝑙−1
𝑙 ], the contrapositive of the Zoom
Lemma tells us that the coefficient 𝑝 𝑑∗ of 𝑝 vanishes at the point (4.1) with 𝑧 = 1.
The multilinear monomials 𝑚 of degree 𝑙 + 1 that match 𝑑 ∗ have the form 𝑚 = 𝑥 𝑖 · 𝑚 ∗ , where
𝑖 ∈ [𝑛] \ 𝐿. Thus, we can write the coefficient 𝑝 𝑑∗ as
Õ Õ
𝑝 𝑑∗ = 𝐶({𝑖} ∪ 𝐼, 𝐽) · 𝑥 𝑖 + 𝐶(𝐼, {𝑖} ∪ 𝐽) · 𝑥 𝑖 . (8.2)
𝑖∈𝑈\𝐼 𝑖∈𝑉\𝐽
For each 𝑖 ∈ [𝑛] \ 𝐿, (4.1) with 𝑧 = 1 substitutes 1/(𝑎 𝑖 − 𝑎 𝑗 ∗ ) into 𝑥 𝑖 . Plugging this into (8.2) yields
(8.1).
Proof of Lemma 8.5. The proof goes by induction on 𝑡. The base case is 𝑡 = 0, where the lemma
holds because the rank of a nonzero matrix is always at least 1. For the inductive step, where
𝑡 ≥ 1, we zoom in on the contributions of the monomials that contain a particular variable. More
precisely, for 𝑗 ∗ ∈ [𝑛], let 𝑝 𝑗 ∗ denote the partial derivative 𝑝 𝑗 ∗ 𝜕𝑥 𝑗∗ 𝑝. Consider any 𝑗 ∗ ∈ [𝑛] such
that 𝑝 𝑗 ∗ is nonzero. As 𝑝 is multilinear and homogeneous of degree 𝑙 + 1, 𝑝 𝑗 ∗ is multilinear and
homogeneous of degree 𝑙. As every monomial in 𝑝 depends on at least 𝑡 variables indexed by 𝑈
and at least 𝑡 variables indexed by 𝑉, every monomial in 𝑝 𝑗 ∗ depends on at least 𝑡 − 1 variables
indexed by 𝑈 and at least 𝑡 − 1 variables indexed by 𝑉. In a moment, we argue that for every
𝑗 ∗ ∈ [𝑛], 𝑝 𝑗 ∗ ∈ Van[RFE𝑙−2
𝑙−1 ]. Then we will show the following:
Claim 8.9. There exists 𝑗 ∗ ∈ [𝑛] such that 𝑝 𝑗 ∗ ≠ 0 and
rank(CMat𝑈 ,𝑉 (𝑝)) ≥ rank(CMat𝑈 ,𝑉 (𝑝 𝑗 ∗ )) + 1. (8.3)
Given 𝑗 ∗ as in Claim 8.9, we conclude by induction that
rank(CMat𝑈 ,𝑉 (𝑝)) ≥ rank(CMat𝑈 ,𝑉 (𝑝 𝑗 ∗ )) + 1 ≥ (𝑡 − 1) + 1 + 1 = 𝑡 + 1.
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To see that 𝑝 𝑗 ∗ belongs to the vanishing ideal of RFE𝑙−2 𝑙−1 , we use Theorem 1.8. Note that 𝑝 𝑗 is
∗
homogeneous, just like 𝑝. Condition 1 of Theorem 1.8 is satisfied by 𝑝 𝑗 ∗ since it it is satisfied
by 𝑝, and all of 𝑘, 𝑙, 𝑛, and the degree of 𝑝 𝑗 ∗ are one less. Given 𝐾 and 𝐿 as in condition 2 of
Theorem 1.8, we have
𝜕𝐿 𝑝 𝑗 ∗ 𝐾←0 = 𝑝 𝑑∗ (8.4)
where 𝑑 ∗ is the degree pattern with domain 𝐾 ∪ 𝐿 ∪ {𝑗 ∗ } that has 𝑑 ∗𝑗 = 1 for 𝑗 ∈ 𝐿 ∪ {𝑗 ∗ } and
𝑑 ∗𝑗 = 0 for 𝑗 ∈ 𝐾. Since 𝑝 ∈ Van[RFE𝑙−1𝑙 ], the contrapositive of the Zoom Lemma applied to
𝑝 with 𝐾 = 𝐾 ∪ {𝑗 }, 𝐿 = 𝐿 ∪ {𝑗 }, 𝑑 ∗ , says that (8.4) is zero upon the substitution (1.7). So
0 ∗ 0 ∗
𝑝 𝑗 ∗ ∈ Van[RFE𝑙−2
𝑙−1 ] by Theorem 1.8. This concludes the proof of Lemma 8.5 modulo the proof of
Claim 8.9.
Proof of Claim 8.9. Let 𝑈 0 ⊆ 𝑈 be the indices of variables 𝑥 𝑖 such that 𝑝 depends on 𝑥 𝑖 , and
similarly define 𝑉 0 ⊆ 𝑉. We first consider the possibility that (8.3) fails for every 𝑗 ∗ ∈ 𝑉 0. We
show that this can only happen when |𝑉 0 | < |𝑈 0 |. A symmetric argument shows that if (8.3)
fails for all 𝑗 ∗ ∈ 𝑈 0, then it must be that |𝑈 0 | < |𝑉 0 |. As both inequalities cannot simultaneously
occur, this guarantees the existence of the desired 𝑗 ∗ .
Suppose that (8.3) fails for each 𝑗 ∗ ∈ 𝑉 0. Observe that the column of CMat𝑈 ,𝑉 (𝑝 𝑗 ∗ ) correspond-
ing to a monomial 𝑚 equals the column of CMat𝑈 ,𝑉 (𝑥 𝑗 ∗ 𝑝 𝑗 ∗ ) corresponding to the monomial 𝑥 𝑗 ∗ 𝑚;
all other columns of CMat𝑈 ,𝑉 (𝑥 𝑗 ∗ 𝑝 𝑗 ∗ ) are zero. The matrix CMat𝑈 ,𝑉 (𝑥 𝑗 ∗ 𝑝 𝑗 ∗ ) can also be formed
from CMat𝑈 ,𝑉 (𝑝) by zeroing out all the columns indexed by subsets that do not contain 𝑗 ∗
(corresponding to multilinear monomials not involving 𝑥 𝑗 ∗ ). The failure of (8.3) for 𝑗 ∗ implies that
CMat𝑈 ,𝑉 (𝑝 𝑗 ∗ ) has the same rank as CMat𝑈 ,𝑉 (𝑝), which is to say that the columns of CMat𝑈 ,𝑉 (𝑝)
indexed by subsets that contain 𝑗 ∗ span all the columns of CMat𝑈 ,𝑉 (𝑝). Going block by block,
this implies that for every block 𝐶 𝑑 of 𝐶 = CMat𝑈 ,𝑉 (𝑝), the columns within 𝐶 𝑑 that are indexed
by subsets containing 𝑗 ∗ span all the columns of 𝐶 𝑑 . This goes for every 𝑗 ∗ ∈ 𝑉 0, as we are
assuming that (8.3) fails for all of them.
Let 𝑑 be minimal such that 𝐶 𝑑 ≠ 0, i. e., such that 𝑝 has a monomial depending on exactly 𝑑
variables indexed by 𝑈. We have 𝑑 ≥ 𝑡 ≥ 1 and 𝐶 𝑑−1 = 0. The entries of 𝐶 𝑑 appear in the linear
equations (8.1) given in Proposition 8.8, either with entries from 𝐶 𝑑−1 or from 𝐶 𝑑+1 . Since 𝐶 𝑑−1
is zero, the equations involving 𝐶 𝑑−1 and 𝐶 𝑑 simplify to equations on 𝐶 𝑑 only. Namely, for every
𝐼 ⊆ 𝑈 with |𝐼 | = 𝑑 − 1, every 𝐽 ⊆ 𝑉 with |𝐽 | = 𝑙 − (𝑑 − 1), and every 𝑗 ∗ ∈ 𝐼 ∪ 𝐽, equation (8.1)
simplifies to
Õ 𝐶 ({𝑖} ∪ 𝐼, 𝐽)
𝑑
= 0. (8.5)
𝑎 𝑖 − 𝑎 𝑗∗
𝑖∈𝑈\𝐼
For any fixed 𝑖 ∈ 𝑈 \ 𝑈 0, all entries of the form 𝐶 𝑑 ({𝑖} ∪ 𝐼, 𝐽) are zero. Thus, we can restrict the
range of 𝑖 in (8.5) from 𝑈 \ 𝐼 to 𝑈 0 \ 𝐼:
Õ 𝐶 ({𝑖} ∪ 𝐼, 𝐽)
𝑑
= 0. (8.6)
𝑎 𝑖 − 𝑎 𝑗∗
𝑖∈𝑈 0 \𝐼
Since 𝐶 𝑑 ≠ 0, there is at least one fixed 𝐼 for which not all entries of the form 𝐶 𝑑 ({𝑖} ∪ 𝐼, 𝐽)
are zero as 𝑖 and 𝐽 vary. Let 𝐼 ∗ be such an 𝐼, and let 𝐶 𝑑∗ denote the submatrix of 𝐶 𝑑 that consists
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of all entries of the form 𝐶 𝑑 ({𝑖} ∪ 𝐼 ∗ , 𝐽) as 𝑖 and 𝐽 vary. For every 𝐽 ⊆ 𝑉 with |𝐽 | = 𝑙 − (𝑑 − 1) and
every 𝑗 ∗ ∈ 𝐼 ∗ ∪ 𝐽, we have
Õ 𝐶 ∗ ({𝑖} ∪ 𝐼 ∗ , 𝐽)
𝑑
= 0. (8.7)
𝑎 𝑖 − 𝑎 𝑗∗
𝑖∈𝑈 \𝐼
0 ∗
For each 𝑗 ∗ ∈ 𝑉 0, consider the equations (8.7) where 𝐽 ranges over all subsets of 𝑉 of size
|𝐽 | = 𝑙 − (𝑑 − 1) that contain 𝑗 ∗ . Observe that the coefficients 𝑎 𝑖 −𝑎
1
𝑗∗
in (8.7) are independent of
the choice of 𝐽. We argued that the columns of 𝐶 𝑑 indexed by subsets 𝐽 that contain 𝑗 ∗ span all
columns of 𝐶 𝑑 . The same holds for 𝐶 𝑑∗ , as 𝐶 𝑑∗ is obtained from 𝐶 𝑑 by removing rows. It follows
that (8.7) holds for every subset 𝐽 of 𝑉 of size 𝑙 − (𝑑 − 1) (not just the ones containing 𝑗 ∗ ).
In particular, consider any one nonzero column of 𝐶 𝑑∗ . The column represents a nontrivial
solution to the homogeneous system (8.7) of |𝑉 0 | linear equations (one for each choice of 𝑗 ∗ ∈ 𝑉 0)
in |𝑈 0 \ 𝐼 ∗ | unknowns (one for each 𝑖 ∈ 𝑈 0 \ 𝐼 ∗ ). The coefficient matrix [ 𝑎 𝑖 −𝑎
1
𝑗∗
] is a Cauchy matrix,
which is well-known to have full rank. In order for there to be a nontrivial solution, the number
of equations must be strictly less than the number of unknowns. In other words, we have
|𝑉 0 | < |𝑈 0 \ 𝐼 ∗ | ≤ |𝑈 0 |, as desired.
9 Alternating Algebra Representation
In this section we present in greater detail the alternating algebra-based representation of
(multilinear) polynomials suited to studying the vanishing ideal of RFE. Subsection 9.1 expands
the informal discussion from the overview, describing the representation and characterization
for the setting when 𝑙 = 1, 𝑘 = 0, and degree 𝑑 = 2. Subsection 9.2 provides a brief introduction
to alternating algebra suited to our purpose. Subsection 9.3 formalizes the discussion from
Subsection 9.1 and extends it to the case of multilinear polynomials for general 𝑘, 𝑙, and 𝑑.
9.1 Basic case
For the purposes of this subsection, we fix the parameters 𝑘 = 0, 𝑙 = 1, and 𝑑 = 2. That is to say,
we are studying which degree-2 polynomials belong to the vanishing ideal for RFE01 .
In Theorem 1.3, we proved that the polynomials EVC01 [𝑖1 , 𝑖2 , 𝑖3 ] as 𝑖1 , 𝑖2 , 𝑖3 range over [𝑛]
generate Van[RFE01 ]. As these generators are all homogeneous degree-2 polynomials, a degree-2
polynomial 𝑝 is in the ideal if and only if it is a linear combination of instantiations of EVC01 .
Consider the generator when expanded as a linear combination of monomials:
𝑎 𝑖1 𝑎 𝑎
𝑥 𝑖1 𝑥 𝑖2 + 𝑖3 𝑥 𝑖3 𝑥 𝑖1 + 𝑖2
1 1 1
EVC01 [𝑖1 , 𝑖2 , 𝑖3 ] = 𝑥 𝑥 .
𝑎 𝑖2 1 𝑎 𝑖1 1 𝑎 𝑖3 1 𝑖2 𝑖3
We represent it graphically by creating a vertex 𝑣 𝑖 ∈ 𝑉 for each variable 𝑥 𝑖 , an undirected
edge for each monomial, and assigning to each edge a weight equal to the coefficient of that
monomial:
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𝑎 𝑖1 1 𝑣 𝑖2 𝑎 𝑖2 1
𝑎 𝑖2 1 𝑎 𝑖3 1
𝑣 𝑖1 𝑣 𝑖3
𝑎 𝑖3 1
𝑎 𝑖1 1
Observe that the coefficient of 𝑥 𝑖1 𝑥 𝑖2 has no dependence on 𝑎 𝑖3 . In particular, as 𝑖3 varies,
the coefficient of 𝑥 𝑖1 𝑥 𝑖2 in EVC01 [𝑖1 , 𝑖2 , 𝑖3 ] does not change. In any other instantiation of EVC01
involving both 𝑖1 and 𝑖2 , the coefficient is either the same, or else differs by a sign, according to
whether 𝑖1 or 𝑖2 precedes the other in the determinant. A similar pattern holds with respect to
all other monomials. This suggests we can modify the graphical representation by rescaling
the weights on edges and suppress the dependence on the abscissas. To capture the signs,
we use oriented edges. More precisely, for each edge {𝑣 𝑖1 , 𝑣 𝑖2 }, we consider either of its two
𝑎
orientations, say 𝑣 𝑖1 → 𝑣 𝑖2 , and then divide its coefficient by 𝑖1
1
. Note that considering the
𝑎 𝑖2 1
opposite orientation coincides with flipping the sign of the scaling factor. With these changes,
EVC01 [𝑖1 , 𝑖2 , 𝑖3 ] may be drawn in any of the following ways (among others).
𝑣 𝑖2 𝑣 𝑖2 𝑣 𝑖2 𝑣 𝑖2
1 1 −1 1 −1 −1 −1 −1
𝑣 𝑖1 𝑣 𝑖3 𝑣 𝑖1 𝑣 𝑖3 𝑣 𝑖1 𝑣 𝑖3 𝑣 𝑖1 𝑣 𝑖3
1 1 1 −1
While different choices of edge orientations lead to different illustrations, any one illustration
can be transformed into any other by considering edges in opposite orientations as needed, and
flipping the sign of each associated coefficient. By identifying each edge in one orientation with
the negative of itself in the opposite orientation, we can view all the illustrations as renditions of
the same underlying object.
In general, we can represent any degree-2 homogeneous multilinear polynomial 𝑝 ∈
𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] in a similar way: For each monomial 𝑥 𝑖1 𝑥 𝑖2 create an oriented edge 𝑣 𝑖1 → 𝑣 𝑖2
𝑎
and set the weight of the edge to be the coefficient of 𝑥 𝑖1 𝑥 𝑖2 in 𝑝 divided by 𝑖1
1
. The
𝑎 𝑖2 1
representation determines the polynomial: simply undo the scaling on each edge, and read off a
linear combination of monomials. Note moreover that this graphical representation is linear in
the polynomial: adding or rescaling polynomials coincides with adding or rescaling coefficients
on the edges.
Observe that, in every graphical representation of EVC01 [𝑖1 , 𝑖2 , 𝑖3 ], at every vertex, the sum of
the coefficients on edges oriented out of that vertex equals the sum of the coefficients on edges
oriented in to that vertex. Indeed, we can interpret EVC01 [𝑖1 , 𝑖2 , 𝑖3 ] as a circulation in which one
unit of flow travels around a simple 3-cycle 𝑣 𝑖1 → 𝑣 𝑖2 → 𝑣 𝑖3 → 𝑣 𝑖1 . The coefficient on an oriented
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edge 𝑣 1 → 𝑣 2 measures how much flow is traveling in the direction 𝑣 1 → 𝑣 2 , with negatives
representing flow in the opposite direction. That the sum of coefficients on outgoing edges
equals the sum of coefficients on incoming edges reflects the defining property of a circulation,
namely that the conservation law holds at every vertex: the total flow in equals the total flow out.
Conservation is maintained under linear combinations. Since every degree-2 polynomial
𝑝 in Van[RFE01 ] is a linear combination of instantiations of EVC01 , the representation of 𝑝 also
satisfies the conservation law at every vertex, i. e., the representation of 𝑝 is a circulation. Thus,
conservation is a necessary condition for membership in Van[RFE01 ].
Conservation is sufficient for ideal membership, as well. By definition, conservation at every
vertex means that the representation is a circulation. By the well-known flow decomposition
theorem (see, e. g., [5, p. 80-81]), every circulation can be decomposed into a superposition of
circulations around simple cycles. A unit circulation around a simple cycle can be decomposed
into a sum of unit circulations around 3-cycles; this is depicted for a 5-cycle below, where each
edge indicates unit flow:
𝑣2 𝑣2 𝑣2
𝑣3 𝑣3 𝑣3
𝑣1 = 𝑣1 = 𝑣1
𝑣4 𝑣4 𝑣4
𝑣5 𝑣5 𝑣5
The basis of the first equality in the above figure is that a unit flow 𝑣 1 → 𝑣 3 cancels with a unit
flow 𝑣 3 → 𝑣 1 , and similar for 𝑣 4 in lieu of 𝑣3 . Thus, conservation implies that we have a linear
combination of unit circulations on 3-cycles, i. e., a linear combination of instantiations of EVC01 .
In summary, a multilinear homogeneous degree-2 polynomial is in Van[RFE01 ] if and only if its
graphical representation satisfies the conservation law at every vertex. This is the representation
and ideal membership characterization in the basic setting with 𝑘 = 0, 𝑙 = 1, and 𝑑 = 2
for multilinear homogeneous polynomials. Note that, in this basic setting, the multilinear
homogeneous degree-2 case represents the core of the problem. The remaining cases contain a
univariate monomial, and are outside of RFE01 by Proposition 5.1.
9.2 Alternating algebra
In order to generalize Subsection 9.1, we need to be able to discuss higher-dimensional analogues
of “flow” and “circulation”, as well as appropriately-generalized notions of “conservation.”
Suited to this purpose is the language of alternating algebra. Alternating algebra was introduced
in the 1800s by Hermann Grassmann [22, 23] and is the formalism underlying differential
geometry and its applications to physics. We give a brief introduction to alternating algebra
here, tailored toward our purposes.
For each 𝑖 ∈ [𝑛], we create a fresh vertex 𝑣 𝑖 ∈ 𝑉, which corresponds to the variable 𝑥 𝑖 . The
alternating algebra provides a multiplication, denoted ∧, that can be thought of as a constructor
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to make oriented simplices out of these vertices. For example, the ∧-product of 𝑣 1 with 𝑣2 ,
written 𝑣1 ∧ 𝑣2 , encodes the simplex with vertices 𝑣 1 and 𝑣 2 in a particular orientation; 𝑣 2 ∧ 𝑣 1
encodes the same simplex with the opposite orientation. When 𝑣 1 = 𝑣 2 , 𝑣 1 ∧ 𝑣 2 is defined to
be zero. ∧-multiplication is associative. Rather than being commutative, the ∧-product is
anti-commutative in the sense that 𝑣1 ∧ 𝑣 2 = −𝑣2 ∧ 𝑣1 . In this way the order of the vertices in the
product encodes an orientation. There are only ever two orientations. In a larger product such
as 𝑣1 ∧ 𝑣 2 ∧ 𝑣 3 , we have
𝑣 1 ∧ 𝑣 2 ∧ 𝑣 3 = −𝑣 1 ∧ 𝑣 3 ∧ 𝑣 2
= 𝑣 3 ∧ 𝑣 1 ∧ 𝑣 2 = −𝑣 3 ∧ 𝑣 2 ∧ 𝑣 1
= 𝑣 2 ∧ 𝑣 3 ∧ 𝑣 1 = −𝑣 2 ∧ 𝑣 1 ∧ 𝑣 3 .
In general, permuting the vertices in a ∧-product by an even permutation has no effect, while
permuting by an odd permutation flips the sign. Any ∧-product that uses the same vertex more
than once is zero.
We can formally extend ∧-multiplication to linear combinations of vertices in 𝑉. Denote 𝑈
to be the 𝔽 -vector space with basis 𝑉. The ∧-multiplication extends to 𝑈 by being distributive.
Overall, ∧-multiplication has the following defining properties, for any 𝑢1 , 𝑢2 , 𝑢3 ∈ 𝑈:
• Associativity: 𝑢1 ∧ (𝑢2 ∧ 𝑢3 ) = (𝑢1 ∧ 𝑢2 ) ∧ 𝑢3 .
• Distributivity: 𝑢1 ∧ (𝑢2 + 𝑢3 ) = 𝑢1 ∧ 𝑢2 + 𝑢1 ∧ 𝑢3 .
• Alternation: 𝑢1 ∧ 𝑢1 = 0.
The alternation property implies anti-commutativity5: 𝑢1 ∧𝑢2 = −𝑢2 ∧𝑢1 . The alternating algebra
consists of all formal linear combinations of ∧-products of vertices from 𝑉, or equivalently, of
elements from 𝑈. We denote the underlying universe as follows.
Definition 9.1 (space of oriented simplices). For each 𝑡 ∈ ℕ , we let
Λ𝑡 (𝑈) span(𝑢1 ∧ · · · ∧ 𝑢𝑡 : 𝑢1 , . . . , 𝑢𝑡 ∈ 𝑈)
denote the space of linear combinations of 𝑡-vertex oriented simplices. For a set of indices 𝑇, we
write 𝑢 𝑇 𝑖∈𝑇 𝑢𝑖 , with the convention that the indices are listed in increasing order.
Ó
The distributivity and anti-commutativity properties of ∧ imply that
Λ𝑡 (𝑈) = span(𝑢 [𝑡] : 𝑢1 , . . . , 𝑢𝑡 are distinct elements in 𝑉),
which justifies the reference to 𝑡-vertex simplices. The properties also imply that changing the
order of the vertices in the wedge product yields the same element up to a sign, namely the
sign of the underlying permutation. This justifies the reference to orientation, where there are
two possible orientations. To emphasize, the 𝑡 in Λ𝑡 (𝑈) counts the number of vertices in the
simplices; this is one more than the usual notion of dimension of a simplex. For 𝑡 = 0, we have
a distinct simplex corresponding to the empty product, denoted 1, which is an identity for ∧.
Note that not every element in Λ𝑡 (𝑈) can be expressed in the form 𝑢1 ∧ · · · ∧ 𝑢𝑡 .
To connect this with Subsection 9.1, recall the graphical depiction of EVC01 [𝑖1 , 𝑖2 , 𝑖3 ]:
5Alternation and anti-commutativity are equivalent provided the characteristic of the field differs from 2.
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𝑣 𝑖2
1 1
𝑣 𝑖1 𝑣 𝑖3
1
Adopting the convention that an arrow 𝑣 1 → 𝑣 2 is 𝑣 1 ∧ 𝑣 2 (and so an arrow 𝑣 2 → 𝑣 1 is
𝑣 2 ∧ 𝑣 1 = −𝑣 1 ∧ 𝑣 2 ), we can alternatively express the above as
𝑣 𝑖1 ∧ 𝑣 𝑖2 + 𝑣 𝑖2 ∧ 𝑣 𝑖3 + 𝑣 𝑖3 ∧ 𝑣 𝑖1 .
In general, the graphical representation of a homogeneous degree-2 multilinear polynomial
is some linear combination of 2-vertex oriented simplices. When we go to higher-degree
polynomials, we make use of oriented simplices with more vertices.
To express conservation, we introduce boundary maps, which are parametrized by a linear
weight function 𝑤 : 𝑈 → 𝔽 . The boundary map 𝜕𝑤 is a linear map that sends each simplex to a
linear combination of its boundary faces (and the empty simplex to zero) according to a formula
reminiscent of the minor expansion of a determinant along a column consisting of the values of
𝑤.
Definition 9.2 (boundary map). For anyÉ 𝑤 : 𝑈 → 𝔽 , the boundary map with
linear functionÉ
𝑛 𝑡 𝑛 𝑡
weight function 𝑤 is the linear map 𝜕𝑤 : 𝑡=0 Λ (𝑈) → 𝑡=0 Λ (𝑈) realizing
𝑡
Õ
𝑢1 ∧ · · · ∧ 𝑢𝑡 ↦→ (−1)𝑖+1 𝑤(𝑢𝑖 )(𝑢1 ∧ · · · ∧ 𝑢𝑖−1 ∧ 𝑢𝑖+1 ∧ · · · ∧ 𝑢𝑡 ) (9.1)
𝑖=1
for all 𝑢1 , . . . , 𝑢𝑡 ∈ 𝑈.
The boundary map 𝜕𝑤 is well-defined. To see this, note that the sign factor (−1)𝑖+1 in (9.1)
ensures well-definedness of the restriction to vertices, i. e., for 𝑢1 , . . . , 𝑢𝑡 ∈ 𝑉. This is because
changing the order of the vertices on the left-hand side results in the correct sign change on the
right-hand side. The linearity of 𝑤 then guarantees that the linear extension of the restriction to
vertices coincides with (9.1). For each 𝑡 ≥ 1, 𝜕𝑤 (Λ𝑡 (𝑈)) ⊆ Λ𝑡−1 (𝑈), while 𝜕𝑤 (Λ0 (𝑈)) = {0}.
In the simplest case, 𝑤 is the function that is 1 on every 𝑣 ∈ 𝑉. In this case, the boundary of
some 2-vertex simplex is given by
𝜕1 (𝑣 1 ∧ 𝑣2 ) = 𝑣 2 − 𝑣1 .
In particular, 𝑣 1 ∧ 𝑣 2 contributes −1 toward 𝑣 1 and +1 toward 𝑣 2 . This coincides with the
contribution of the edge 𝑣1 → 𝑣 2 toward the net flow into the vertices 𝑣1 and 𝑣 2 . In exactly this
way, conservation is identified with having a vanishing boundary. Note also that for this choice of
weight function
𝜕1 (𝑣 1 ∧ 𝑣2 ∧ 𝑣3 ) = 𝑣 2 ∧ 𝑣3 − 𝑣 1 ∧ 𝑣 3 + 𝑣 1 ∧ 𝑣 2 = 𝑣 1 ∧ 𝑣 2 + 𝑣 2 ∧ 𝑣 3 + 𝑣 3 ∧ 𝑣 1 .
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Thus, unit circulations on 3-cycles are in one-to-one and onto correspondence with the images
under 𝜕1 of oriented 3-simplices on the vertices. By the decomposition discussed in the
Subsection 9.1, it follows that circulations are in one-to-one and onto correspondence with the
elements of 𝜕1 (Λ3 (𝑈)). This means that 𝜕1 (Λ3 (𝑈)) = ker(𝜕1 ) ∩ Λ2 (𝑈).
In general, for every linear 𝑤 : 𝑈 → 𝔽
im(𝜕𝑤 ) = ker(𝜕𝑤 ), (9.2)
or equivalently, 𝜕𝑤 (Λ𝑡 (𝑈)) = ker(𝜕𝑤 ) ∩ Λ𝑡−1 (𝑈) for every 𝑡 ∈ [𝑛]. This key relationship
implies that taking the same boundary multiple times always vanishes. That is, for any 𝑤,
𝜕𝑤 ◦ 𝜕𝑤 = 0, often written as 𝜕𝑤 2 = 0. Another property is that for any 𝑤, 𝑤 0 and 𝛽, 𝛽0 ∈ 𝔽 ,
𝜕𝛽𝑤+𝛽0 𝑤0 = 𝛽𝜕𝑤 + 𝛽 𝜕𝑤0 , which is to say that the boundary maps themselves are linear in 𝑤. It
0
follows from these that, for any 𝑤, 𝑤 0, 𝜕𝑤 ◦ 𝜕𝑤0 = −𝜕𝑤0 ◦ 𝜕𝑤 . This means that the boundary maps
themselves behave like an alternating algebra, with ◦ as the multiplication rather than ∧. For any
𝑤 1 , . . . , 𝑤 𝑘+1 , write 𝜔 = 𝑤1 ∧ · · · ∧ 𝑤 𝑘+1 , and define 𝜕𝜔 = 𝜕𝑤 𝑘+1 ◦ · · · ◦ 𝜕𝑤1 . That is, 𝑤1 ∧ · · · ∧ 𝑤 𝑘+1
means apply 𝜕𝑤1 , then 𝜕𝑤2 , and so on, up to 𝜕𝑤 𝑘+1 . The result is well-defined, and we borrow
the shorthand notation introduced in Definition 9.1: 𝑤 𝑇 𝑗∈𝑇 𝑤 𝑗 , where 𝑇 ⊆ [𝑘 + 1] and the
Ó
indices in the wedge product are taken in increasing order.
The image-kernel relationship (9.2) extends as follows: For any linearly independent
𝑤 1 , . . . , 𝑤 𝑘+1 ,
𝑘+1
Ù
im(𝜕𝑤1 ∧ · · · ∧ 𝑤 𝑘+1 ) = ker(𝜕𝑤 𝑟 ). (9.3)
𝑟=1
If 𝑤1 , . . . 𝑤 𝑘+1 are linearly dependent, then 𝜕𝑤1 ∧ · · · ∧ 𝑤 𝑘+1 vanishes. In fact, a further generalization
holds and will be useful. We include a proof for completeness. (9.3) corresponds to the special
case Δ = 0.
Proposition 9.3 (generalized image-kernel relationship). For 𝑘, Δ ∈ ℕ and any linearly independent
linear functions 𝑤 1 , . . . , 𝑤 𝑘+Δ+1 : 𝑈 → 𝔽
Ù
span im(𝜕𝑤 𝐵 ) = ker(𝜕𝑤 𝐵 ). (9.4)
𝐵⊆[𝑘+Δ+1] 𝐵⊆[𝑘+Δ+1]
|𝐵|=𝑘+1 |𝐵|=Δ+1
Proof. Extend 𝑤 1 , . . . , 𝑤 𝑘+Δ+1 to a basis 𝑤1 , . . . , 𝑤 𝑛 of all linear functions 𝑈 → 𝔽 . We can
interpret 𝑤 1 , . . . , 𝑤 𝑛 as a basis of the dual space 𝑈 ∗ , and the mapping (𝑤, 𝑢) ↦→ 𝑤(𝑢) as a bilinear
form 𝑈 ∗ × 𝑈 → 𝔽 . This means we can construct a dual basis 𝑢1 , . . . , 𝑢𝑛 ∈ 𝑈 such that for
𝑖, 𝑗 ∈ [𝑛], 𝑤 𝑗 (𝑢𝑖 ) is 1 if 𝑖 = 𝑗 and 0 if 𝑖 ≠ 𝑗.
In this particular basis 𝑢1 , . . . , 𝑢𝑛 , the boundary maps with weight functions 𝑤 𝑗 take a very
simple form: The only term in (9.1) that remains for 𝑤 = 𝑤 𝑗 is the one with 𝑖 = 𝑗. More generally,
for 𝐵, 𝑇 ⊆ [𝑛], (
±𝑢 𝑇\𝐵 if 𝐵 ⊆ 𝑇
𝜕𝑤 𝐵 (𝑢 𝑇 ) = (9.5)
0 otherwise.
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With this characterization, we can see that both the span of the images and the intersection of
the kernels coincide with
span(𝑢 𝑆 : 𝑆 ⊆ [𝑛] with |𝑆 ∩ [𝑘 + Δ + 1]| ≤ Δ).
We obtain ±𝑢 𝑆 as 𝜕𝑤 𝐵 (𝑢 𝑇 ) if and only if 𝑇 = 𝐵t𝑆. Such a choice of 𝑇 and 𝐵 with 𝐵 ⊆ [𝑘 +Δ+1]
and |𝐵| = 𝑘 + 1 exists if and only if there are at least 𝑘 + 1 elements in [𝑘 + Δ + 1] \ 𝑆, or equivalently,
|𝑆 ∩ [𝑘 + Δ + 1]| ≤ (𝑘 + Δ + 1) − (𝑘 + 1) = Δ. This proves the equality for the span of the images.
On the other hand, 𝑢 𝑆 falls within ker(𝜕𝑤 𝐵 ) if and only if 𝐵 * 𝑆. This is case for every
𝐵 ⊆ [𝑘 + Δ + 1] with |𝐵| = Δ + 1 if and only if 𝑆 contains at most Δ elements in [𝑘 + Δ + 1]. This
proves the equality of the intersection of the kernels.
In the following subsection, we will need an explicit formula for computing 𝜕𝜔 (𝑢 𝑇 ) for
generic 𝑢1 , . . . , 𝑢𝑛 ∈ 𝑈 and 𝑇 ⊆ [𝑛]. From a concrete perspective, the effect of a single boundary
map in Definition 9.2 resembles one level of determinant minor expansion, so composing
boundary maps should produce a partially expanded determinant. We formalize that intuition
with the following proposition, which characterizes the boundary of a 𝑡-simplex after applying
𝑘 weighted boundaries as a linear combination of (𝑡 − 𝑘)-simplices. Each (𝑡 − 𝑘)-simplex is
indexed by a subset 𝐽 of 𝑇.
Proposition 9.4 (composed boundary maps). Let 𝑤 1 , . . . , 𝑤 𝑘+1 : 𝑈 → 𝔽 be linear functions, 𝑇 a set
of indices, and 𝑢𝑖 ∈ 𝑈 for 𝑖 ∈ 𝑇.
Õ 𝑟∈[𝑘+1]
𝜕𝑤 [𝑘+1] (𝑢 𝑇 ) = (−1)XInv(𝐼,𝐽) · det 𝑤 𝑟 (𝑢𝑖 ) 𝑖∈𝐼 · 𝑢𝐽 ,
(9.6)
𝐼t𝐽=𝑇
|𝐼 |=𝑘+1
where in the determinant, the rows from top to bottom and the columns from left to right are in increasing
order of index 𝑖 and 𝑟, respectively.
Proof. Observe that 𝜕𝑤 [𝑘+1] (𝑢 𝑇 ) ∈ Λ𝑡−𝑘−1 (𝑈) and, by Definition 9.2, can be written as a linear
combination of 𝑢 𝐽 over all 𝐽 ⊆ 𝑇 with |𝐽 | = |𝑇 | − 𝑘 − 1. It suffices to show that the coefficients of
each 𝑢 𝐽 match the ones given above.
Without loss of generality, let 𝑇 = [𝑡], as this does not change the relative order of any
determinants or ∧-products. Consider the terms formed by iteratively expanding 𝜕𝑤 [𝑘+1] (𝑢 𝑇 )
𝜕𝑤1 ∧ · · · ∧ 𝑤 𝑘+1 (𝑢 𝑇 ) by Definition 9.2. Each term is in one-to-one correspondence with the choices
of 𝑖 we make in the expansions of Definition 9.2. In particular, the terms that yield 𝑢 𝐽 correspond
to the bijections 𝜎 : [𝑘 + 1] → 𝐼, where 𝐼 = 𝑇 \ 𝐽. For a given 𝜎, the corresponding coefficient is
equal to Ö
(−1)( 𝑖∈𝐼 𝑖)+𝑘+1−|{𝑟,𝑟 ∈[𝑘+1]:𝑟 <𝑟,𝜎(𝑟 )<𝜎(𝑟)}|
Í 0 0 0
𝑤 𝑟 (𝑢𝜎(𝑟) ).
𝑟∈[𝑘+1]
The |{𝑟 0 < 𝑟, 𝜎(𝑟 0) < 𝜎(𝑟)}| term accounts for the fact that, when each 𝑟 is selected, some terms
of 𝑇 may have been previously removed, shifting the relative rank of 𝑟. Since for any distinct
𝑟, 𝑟 0 ∈ [𝑘 + 1], either 𝜎(𝑟 0) > 𝜎(𝑟) or 𝜎(𝑟 0) < 𝜎(𝑟), we can rewrite |{𝑟 0 < 𝑟, 𝜎(𝑟 0) < 𝜎(𝑟)}| =
𝑘+1
2 − |{𝑟 < 𝑟, 𝜎(𝑟 ) > 𝜎(𝑟)}|.
0 0
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As for the term 𝑖∈𝐼 𝑖, writing 𝑖 as 𝑖 = |{𝑖 0 ∈ 𝐼 : 𝑖 0 ≤ 𝑖}| + |{𝑗 ∈ 𝐽 : 𝑗 < 𝑖}| and summing over
Í
𝑟 + XInv(𝐼, 𝐽) = 𝑘+2
Í 𝑘+1
all 𝑖 ∈ 𝐼, we have that 𝑖∈𝐼 𝑖 = 𝑟=1 + XInv(𝐼, 𝐽).
Í
2
𝑘+2 𝑘+1
As 2 = 2 + 𝑘 + 1, we get that the coefficient of 𝑢 𝐽 equals
Õ Ö
(−1)XInv(𝐼,𝐽)+|{𝑟 <𝑟,𝜎(𝑟 )>𝜎(𝑟)}|+2𝑘+2
0 0
𝑤 𝑟 (𝑢𝜎(𝑟) ),
𝜎:[𝑘+1]→𝐼 𝑟∈[𝑘+1]
and by definition of the determinant and simplifying, this is equal to
𝑟∈[𝑘+1]
(−1)XInv(𝐼,𝐽) det 𝑤 𝑟 (𝑢𝑖 ) 𝑖∈𝐼 .
In the next subsection we will apply Proposition 9.4 with 𝑢𝑖 = 𝑣 𝑖 . For the choice of 𝑢𝑖 in the
proof of Proposition 9.3, the matrix in (9.6) is the identity matrix and thus has determinant 1,
which results in (9.5).
9.3 General case
With the notation of alternating algebra in hand, we turn now to generalizing the characterization
of Van[RFE𝑙𝑘 ] based on the representation of polynomials that we introduced in Subsection 9.1,
henceforth the simplicial representation. We focus on multilinear polynomials, but the parameters
𝑘, 𝑙 ∈ ℕ may be arbitrary. For starters, we still restrict to degree 𝑑 = 𝑙 + 1. We then generalize to
multilinear polynomials of arbitrary degree and present an alternate proof to Theorem 1.8. We
end with some thoughts about the non-multilinear case.
As before, we associate each variable 𝑥 𝑖 with a distinct vertex 𝑣 𝑖 ∈ 𝑉, where 𝑈 span(𝑉)
denotes an underlying vector space over 𝔽 . We view a polynomial as a linear combination of
monomials and represent each degree-𝑡 multilinear monomial as an oriented simplex with 𝑡
vertices. The representation makes use of the Vandermonde determinants det(𝐴𝑇 ) for 𝑇 ⊆ [𝑛],
where 𝐴𝑇 refers to the notation that we introduced in (3.2) for the Vandermonde matrix built
from the abscissas 𝑎 𝑖 for 𝑖 ∈ 𝑇 in increasing order. The Vandermonde determinant det(𝐴𝑇 ) can
be written as the product of pairwise differences:
Ö
det(𝐴𝑇 ) = (𝑎 𝑖 − 𝑎 𝑗 ). (9.7)
𝑖,𝑗∈𝑇,𝑖<𝑗
In particular, as the abscissas are distinct, det(𝐴𝑇 ) is always nonzero.
Let 𝑣 𝑇 𝑖∈𝑇 𝑣 𝑖 , where the indices are listed in increasing order. We represent the monomial
Ó
𝑥 𝑇 𝑖∈𝑇 𝑥 𝑖 for 𝑇 ⊆ [𝑛] by the element 𝑣 𝑇 /det(𝐴𝑇 ). Formally, we define the following “decoder
Î
map,” which maps a simplicial representation to the polynomial it represents.
É𝑛 𝑡
Definition 9.5 (representation). 𝜌 : 𝑡=0 Λ (𝑈) → 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] is the linear map extending
𝑣 𝑇 ↦→ det(𝐴𝑇 ) · 𝑥 𝑇 (9.8)
for every 𝑇 ⊆ [𝑛].
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Note that (9.8) holds irrespective of the order of the indices, as long as the same order is used
for both 𝑣 𝑇 and det(𝐴𝑇 ). This is because exchanging any two indices changes the sign of both
the left-hand side and the determinant on the right-hand side. The mapping 𝜌 induces a vector
space isomorphism between Λ𝑙+1 (𝑈) and the space of multilinear homogeneous degree-(𝑙 + 1)
polynomials.
The strategy for our membership test in Van[RFE] consists of two steps: First express
Van[RFE] in terms of 𝜌 and the image of the boundary maps 𝜕𝑤 , and then apply the (generalized)
image-kernel relationship from alternating algebra. In Definition 9.2, 𝑤 is taken to be a linear
function from 𝑈 to 𝔽 . As a linear function, 𝑤 is completely defined by its values on the basis
𝑉. By Lagrange interpolation, every function 𝑤 from 𝑉 to 𝔽 can be viewed as a univariate
polynomial of degree less than 𝑛 |𝑉 | restricted to the abscissas, namely the polynomial
interpolating 𝑎 𝑖 ↦→ 𝑤(𝑣 𝑖 ) for 𝑖 ∈ [𝑛].
Definition 9.6 (degree of boundary map). Let 𝑤 : 𝑈 → 𝔽 be linear and 𝑎 1 , . . . , 𝑎 𝑛 be distinct
elements of 𝔽 . We say that 𝑤 is interpolated by 𝑞 ∈ 𝔽 [𝛼] if 𝑤(𝑣 𝑖 ) = 𝑞(𝑎 𝑖 ) for 𝑖 ∈ [𝑛]. We say that 𝑤
is of degree 𝑑 if 𝑤 is interpolated by a degree-𝑑 polynomial 𝑞.
Furthermore, given fixed 𝑎 1 , . . . , 𝑎 𝑛 , the correspondence between a weight function 𝑤 and
its interpolating polynomial 𝑞 forms an isomorphism; if 𝑤 1 , 𝑤2 are interpolated by 𝑞1 , 𝑞2 , then
𝑤 1 + 𝑤 2 is interpolated by 𝑞1 + 𝑞2 , and 𝑐𝑤1 is interpolated by 𝑐𝑞1 . From now on, we directly
refer to a weight function by the polynomial in 𝔽 [𝛼] that interpolates it. We will be interested in
the boundaries that are weighted by low-degree polynomials.
Multilinear case for degree 𝑑 = 𝑙 + 1. In the case of degree 𝑑 = 𝑙 + 1, the first step of our
approach boils down to finding a simplicial representation for the generators EVC𝑙𝑘 . We do so
using composed boundary maps of degree at most 𝑘.
Lemma 9.7. For any 𝑘, 𝑙 ∈ ℕ and 𝑆 ⊆ [𝑛], |𝑆| = 𝑘 + 𝑙 + 2,
EVC𝑙𝑘 [𝑆] = 𝜌 𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼0 𝑣 𝑆 . (9.9)
That is, EVC𝑙𝑘 is the polynomial formed from a given (𝑘 + 𝑙 + 2)-vertex simplex by iteratively
applying to it the 𝑘 + 1 boundaries weighted by 𝛼 𝑘 , 𝛼 𝑘−1 , . . . , 𝛼0 respectively, where 𝛼 𝑟 stands
for the weight function interpolated by the polynomial 𝛼 𝑟 .
Proof. Using our notation, the explicit expression (3.3) in Proposition 3.4 can be rewritten as
Õ
EVC𝑙𝑘 [𝑆] = (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · det(𝐴𝐿 ) · 𝑥 𝐿 . (9.10)
𝐾t𝐿=𝑆
|𝐾|=𝑘+1
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For the right-hand side, we use Proposition 9.4 to get:
Õ
𝜌 𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼 0 𝑣 𝑆 = (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · 𝜌 𝑣 𝐿
𝐾t𝐿=𝑆
|𝐾|=𝑘+1
Õ
= (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · det(𝐴𝐿 ) · 𝑥 𝐿 .
𝐾t𝐿=𝑆
|𝐾|=𝑘+1
The sum is identical to (9.10).
Lemma 9.7 yields the following characterization of the part of Van[RFE𝑙𝑘 ] of degree 𝑙 + 1.
We state it in a format to which we can directly apply the image-kernel relationship (9.3).
Corollary 9.8. For any 𝑘, 𝑙 ∈ ℕ , the set of polynomials of degree 𝑙 + 1 in Van[RFE𝑙𝑘 ] is given by
𝜌(𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼0 (Λ 𝑘+𝑙+2 (𝑈))).
Proof. Since every degree-(𝑙 + 1) polynomial 𝑝 in Van[RFE𝑙𝑘 ] is a linear combination of instantia-
tions of EVC𝑙𝑘 , Lemma 9.7 allows to us to express the subset in Van[RFE𝑙𝑘 ] as
span 𝜌(𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼0 (𝑣 𝑆 )).
𝑆⊆[𝑛]
|𝑆|=𝑘+𝑙+2
The result follows by linearity and the fact that 𝑈 = span(𝑉).
The image-kernel relationship (9.3) then leads to the following membership test. Recall that
𝜌 induces an isomorphism from the space of (𝑙 + 1)-vertex oriented simplices Λ𝑙+1 (𝑈) to the set
of multilinear polynomials of degree 𝑙 + 1, so 𝜌−1 is well-defined on multilinear polynomials.
Theorem 9.9. Let 𝑘, 𝑙 ∈ ℕ . For any multilinear polynomial 𝑝 ∈ 𝔽 [𝑥1 , . . . , 𝑥 𝑛 ] of degree 𝑙 + 1,
𝑝(RFE𝑙𝑘 ) = 0 if and only if 𝑝 is homogeneous of degree 𝑙 + 1 and
𝜕𝑤 (𝜌−1 (𝑝)) = 0
for every weight function 𝑤 of degree at most 𝑘.
Proof. The criterion in Corollary 9.8 can be rewritten as
𝜌−1 (𝑝) ∈ 𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼0 (Λ 𝑘+𝑙+2 (𝑈)).
By Proposition 9.3, this is equivalent to
𝑘
!
Ù
𝜌−1 (𝑝) ∈ ker(𝜕𝛼 𝑟 ) ∩ Λ𝑙+1 (𝑈).
𝑟=0
The intersection with Λ𝑙+1 (𝑈) means that 𝑝 is homogeneous of degree 𝑙 +1. For such polynomials
𝑝, we have that 𝑝(RFE𝑙𝑘 ) = 0 if and only if 𝜕𝛼 𝑟 (𝜌−1 (𝑝)) = 0 for 𝑟 = 0, . . . , 𝑘, which by linearity is
equivalent to 𝜕𝑤 (𝜌−1 (𝑝)) = 0 for all weight functions 𝑤 of degree at most 𝑘.
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Theorem 9.9 states that a multilinear polynomial 𝑝 of degree 𝑙 + 1 is in the vanishing ideal
of RFE𝑙𝑘 if and only if it is homogeneous of degree 𝑙 + 1 and the simplicial representation of 𝑝
satisfies conservation with respect to all degree-𝑘 boundaries. This is the representation and
ideal membership characterization for such polynomials for general 𝑘 and 𝑙 in the special case
of degree 𝑑 = 𝑙 + 1. As we will argue in Proposition 9.14, the characterization coincides with the
membership test from Theorem 1.8 for multilinear polynomials of degree 𝑙 + 1.
In Section 9.1 we considered the special case with 𝑘 = 0 and 𝑙 = 1. In that basic setting,
the only weight functions of degree 𝑘 are the constant functions, and only 𝑤 ≡ 1 needs to be
considered in Theorem 9.9. The resulting criterion is exactly the conservation criterion that we
developed in Section 9.1.
Note that the restriction in Theorem 9.9 to multilinear polynomials 𝑝 is just to ensure that
𝜌−1 (𝑝) is well-defined. For polynomials of degree 𝑙 + 1 that are not multilinear, one could
interpret the non-existence of 𝜌−1 (𝑝) as not satisfying the criterion. This is consistent with
Proposition 5.1, which implies that polynomials of degree 𝑙 + 1 that are not multilinear are
automatically outside Van[RFE𝑙𝑘 ] since they necessarily have a monomial supported on 𝑙 or
fewer variables.
Through Lemma 9.7, the property that 𝜕𝑤 (𝜌−1 (EVC𝑙𝑘 )) = 0 for every weight function 𝑤 of
degree at most 𝑘 can be viewed as an application of 𝜕𝑤∧𝛼 𝑘 ∧ · · · ∧ 𝛼0 = 0 to Λ𝑡 (𝑈) with 𝑡 = 𝑘 + 𝑙 + 2.
The equations (1.6) follow in a similar way from an application with 𝑡 = 𝑘 + 𝑙 + 3.
Multilinear case of arbitrary degree. The two-step approach underlying Theorem 9.9 extends
to multilinear polynomials of higher degrees. Whereas in the special case of degree 𝑑 = 𝑙 + 1
we only needed simplicial representations for EVC𝑙𝑘 [𝑆] in the first step, we now need them for
polynomials of the more general form EVC𝑙𝑘 [𝑆] · 𝑥 𝑀 where 𝑀 ⊆ [𝑛] is disjoint from 𝑆. We can
handle the additional term 𝑥 𝑀 in Lemma 9.7 by including a multiplicative factor
Ö
𝜇𝑀 (𝛼) (𝛼 − 𝑎 𝑗 ) (9.11)
𝑗∈𝑀
in each of the weight functions. The extra factor acts as a masking term and ensures that in the
expansions of (9.1) the terms with 𝑖 ∈ 𝑀 vanish, so under 𝜌 the factor 𝑥 𝑀 remains.
Lemma 9.10. For any 𝑘, 𝑙 ∈ ℕ , 𝑆 t 𝑀 ⊆ [𝑛] with |𝑆| = 𝑘 + 𝑙 + 2, and 𝜇𝑀 (𝛼) 𝑗∈𝑀 (𝛼 − 𝑎 𝑗 ),
Î
det(𝐴𝑆 )
EVC𝑙𝑘 [𝑆] · 𝑥 𝑀 = · 𝜌(𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝑣 𝑆t𝑀 )). (9.12)
det(𝐴𝑆t𝑀 )
Proof. Expand 𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝑣 𝑆t𝑀 ) by Proposition 9.4. Notice that the only nonzero
terms in the expansion correspond to subsets 𝐽 that contain 𝑀. Substituting 𝐼 ← 𝐾 and
𝐽 ← 𝐿 t 𝑀, and factoring out the 𝜇𝑀 (𝑎 𝑖 ) terms from the determinant, we can write
!
Õ Ö
𝑆t𝑀
𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝑣 )= (−1) XInv(𝐾,𝐿t𝑀)
· 𝜇𝑀 (𝑎 𝑖 ) · det(𝐴𝐾 ) · 𝑣 𝐿t𝑀 .
𝐾t𝐿=𝑆 𝑖∈𝐾
|𝐾|=𝑘+1
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Applying 𝜌 yields
!
Õ Ö
𝜌(𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝑣 𝑆t𝑀 )) = (−1)XInv(𝐾,𝐿t𝑀) · 𝜇𝑀 (𝑎 𝑖 ) ·det(𝐴𝐾 )·det(𝐴𝐿t𝑀 )· 𝑥 𝐿t𝑀 .
𝐾t𝐿=𝑆 𝑖∈𝐾
|𝐾|=𝑘+1
(9.13)
Applying (9.7) to 𝑇 = 𝐿 t 𝑀, 𝑇 = 𝐿, and 𝑇 = 𝑀, rearranging terms, and remembering that 𝐴𝑇
takes rows in increasing index, we obtain
© Ö
det(𝐴𝐿t𝑀 ) = (−1)XInv(𝐿,𝑀) · (𝑎 𝑖 − 𝑎 𝑗 )® · det(𝐴𝐿 ) · det(𝐴 𝑀 ).
ª
(9.14)
«𝑖∈𝐿,𝑗∈𝑀 ¬
We can expand (−1)XInv(𝐾,𝐿t𝑀) as the product (−1)XInv(𝐾,𝐿) (−1)XInv(𝐾,𝑀) because XInv(𝐾, 𝐿 t 𝑀)
equals the sum XInv(𝐾, 𝐿) + XInv(𝐾, 𝑀). By the definition of 𝜇𝑀 , we can expand 𝑖∈𝐾 𝜇𝑀 (𝑎 𝑖 ) as
Î
𝑖∈𝐾,𝑗∈𝑀 (𝑎 𝑖 − 𝑎 𝑗 ). Those expansions and (9.14) allow us to write the summand on the right-hand
Î
side of (9.13) as
Ö
(−1)XInv(𝐾,𝐿) (−1)XInv(𝐾,𝑀) (−1)XInv(𝐿,𝑀) · (𝑎 𝑖 − 𝑎 𝑗 )® · det(𝐴𝐾 ) · det(𝐴𝐿 ) · det(𝐴 𝑀 ) · 𝑥 𝐿t𝑀
© ª
«𝑖∈𝐾t𝐿,𝑗∈𝑀 ¬
Using the similar fact as above that (−1)XInv(𝐾t𝐿,𝑀) = (−1)XInv(𝐾,𝑀) (−1)XInv(𝐿,𝑀) , recalling that
𝐾 t 𝐿 = 𝑆, and pulling out the terms independent of the choice of 𝐾, we obtain
𝜌(𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝑣 𝑆t𝑀 ))
© Ö Õ
= (−1)XInv(𝑆,𝑀) · (𝑎 𝑖 − 𝑎 𝑗 )® · det(𝐴 𝑀 ) · 𝑥 𝑀 (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · det(𝐴𝐿 ) · 𝑥 𝐿
ª
«𝑖∈𝑆,𝑗∈𝑀 ¬ 𝐾t𝐿=𝑆
|𝐾|=𝑘+1
det(𝐴𝑆t𝑀 ) 𝑀 Õ
= ·𝑥 (−1)XInv(𝐾,𝐿) · det(𝐴𝐾 ) · det(𝐴𝐿 ) · 𝑥 𝐿 ,
det(𝐴𝑆 )
𝐾t𝐿=𝑆
|𝐾|=𝑘+1
where the last step applies (9.14) with 𝐿 ← 𝑆. By Proposition 3.4, this establishes the result.
The multilinear elements in Van[RFE𝑙𝑘 ] are exactly the linear combinations of terms of the
form (9.12) where 𝑆 ⊆ [𝑛] ranges over subsets of size 𝑘 + 𝑙 + 2 and 𝑀 ⊆ [𝑛] over subsets disjoint
with 𝑆. In order to obtain a simpler characterization of the same type, as well as one to which
we can apply the generalized image-kernel relationship, we show that we can replace the weight
functions on the right-hand side of (9.12) by generic weight functions of the same degree or by
Lagrange interpolants with respect to a subset of abscissas of size one more.
Proposition 9.11. Let 𝑘 + 1, 𝑚, 𝑡 ∈ ℕ with 𝑡 ≥ 𝑘 + 1, 𝜈 ∈ Λ𝑡 (𝑈), and 𝑁 ⊆ [𝑛] with |𝑁 | = 𝑘 + 𝑚 + 1.
Let 𝐿 𝑁 ,𝑗 for 𝑗 ∈ 𝑁 denote the Lagrange interpolants for the subset of abscissas {𝑎 𝑖 } 𝑖∈𝑁 , i. e., 𝐿 𝑁 ,𝑗
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denotes the unique univariate polynomial of degree at most |𝑁 | − 1 satisfying 𝐿 𝑁 ,𝑗 (𝑎 𝑖 ) = 1 for 𝑖 = 𝑗 and
𝐿 𝑁 ,𝑗 (𝑎 𝑖 ) = 0 for 𝑖 ∈ 𝑁 \ {𝑗}. For all weight functions 𝑤 1 , . . . , 𝑤 𝑘+1 of degree at most 𝑘 + 𝑚,
span 𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝜈) = span 𝜕𝑤 [𝑘+1] (𝜈) = span 𝜕𝐿𝐵 (𝜈). (9.15)
𝑁
𝑀⊆𝑁 𝑤1 ,...,𝑤 𝑘+1 ∈𝔽 [𝛼] 𝐵⊆𝑁
|𝑀 |=𝑚 deg(𝑤 1 ),... deg(𝑤 𝑘+1 )≤𝑘+𝑚 |𝐵|=𝑘+1
Some explanation of the compact notation on the right-hand side of (9.15) is in order. First,
we use 𝐿 𝑁 ,𝑗 to differentiate with the notation 𝐿 𝑗 for Lagrange interpolants that we introduced
in Definition 1.1, where 𝐿 𝑗 corresponds to 𝐿[𝑛],𝑗 . Second, for a subset 𝐵 ⊆ 𝑁, we write 𝐿𝐵𝑁 as a
shorthand for 𝑗∈𝐵 𝐿 𝑁 ,𝑗 , where the indices in the wedge product are taken in increasing order.
Ó
Finally, in the composed boundary operator 𝜕𝐿𝐵 , the Lagrange interpolant 𝐿 𝑁 ,𝑗 represents the
𝑁
weight function interpolated by 𝐿 𝑁 ,𝑗 as in Definition 9.6.
Proof. The inclusion ⊆ of the first equality in (9.15) follows because the weight functions 𝜇𝑀 (𝛼)𝛼 𝑟
for 𝑟 ∈ {0, . . . , 𝑘} have degree at most 𝑘 + |𝑀| = 𝑘 + 𝑚.
To argue the inclusion ⊆ of the second equality in (9.15), note that the Lagrange interpolants
𝐿 𝑁 ,𝑗 for 𝑗 ∈ 𝑁 are linearly independent and that there are as many of them as the dimension of
the space of polynomials of degree at most |𝑁 | − 1 = 𝑘 + 𝑚, so they form a basis for that space.
In particular, we can write all weight functions 𝑤 1 , . . . , 𝑤 𝑘+1 of degree at most 𝑘 + 𝑚 as linear
combinations of the Lagrange interpolants 𝐿 𝑁 ,𝑗 , 𝑗 ∈ 𝑁. By the distributivity and antisymmetry
of the wedge product, this implies that
𝜕𝑤 [𝑘+1] (𝜈) ∈ span 𝜕𝐿𝐵 (𝜈).
𝑁
𝐵⊆𝑁
|𝐵|=𝑘+1
It remains to argue that the right-most side of (9.15) is included in the left-most side. Fix
a subset 𝐵 ⊆ 𝑁 of size |𝐵| = 𝑘 + 1. Since the polynomials 𝐿 𝑁 ,𝑗 for 𝑗 ∈ 𝐵 individually have
roots in all but one element of {𝑎 𝑖 } 𝑖∈𝑁 , they collectively have common roots among exactly
|𝑁 | − |𝐵| = 𝑚 of these abscissas, which form a set 𝑀 ⊆ 𝑁. Each 𝐿 𝑁 ,𝑗 can therefore be written as
the product of 𝜇𝑀 and a polynomial of degree at most 𝑘, or equivalently, as a linear combination
of 𝜇𝑀 (𝛼)𝛼 𝑘 , . . . , 𝜇𝑀 (𝛼)𝛼0 . Once again, by the distributivity and antisymmetry of the wedge
product, we have that
𝜕𝐿𝐵 (𝜈) ∈ span 𝜕𝜇𝑀 (𝛼)𝛼 𝑘 ∧ · · · ∧ 𝜇𝑀 (𝛼)𝛼0 (𝜈).
𝑁
𝑀⊆𝑁
|𝑀|=𝑚
The first equality in (9.15) connects weight functions as on the right-hand side of (9.12) with
generic ones of the same degree. This leads to the following simple characterization of the
multilinear part of Van[RFE𝑙𝑘 ] in terms of 𝜌 and the image of composed boundary maps. The
characterization naturally decomposes into separate ones for the homogeneous components of
the various degrees 𝑑.
Corollary 9.12. For any 𝑘, 𝑙 ∈ ℕ , the set of multilinear polynomials 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] in Van[RFE𝑙𝑘 ]
𝑛−𝑘−1
is given by the direct sum ⊕𝑑=0 𝐻𝑑 of homogeneous components of degree 𝑑 ∈ {0, . . . , 𝑛 − 𝑘 − 1} given
by
𝐻𝑑 span 𝜌(𝜕𝑤 [𝑘+1] (Λ 𝑘+𝑑+1 (𝑈))), (9.16)
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where 𝑤1 , . . . , 𝑤 𝑘+1 range over all weight functions of degree at most 𝑘 + 𝑑 − 𝑙 − 1.
For 𝑑 ≤ 𝑙, the only possible choices for the weight functions 𝑤 1 , . . . 𝑤 𝑘+1 in Corollary 9.12
are linearly dependent, which implies that 𝜕𝑤 [𝑘+1] vanishes and therefore 𝐻𝑑 only contains the
zero polynomial. This is consistent with Proposition 5.1, as is the restriction 𝑑 ≤ 𝑛 − 𝑘 − 1.
Proof. By Theorem 1.3 and the fact that all the instantiations EVC𝑙𝑘 are homogeneous of degree
𝑙 + 1, the multilinear elements in Van[RFE𝑙𝑘 ] are exactly the linear combinations of terms of the
form (9.12) where 𝑆 ⊆ [𝑛] ranges over subsets of size 𝑘 + 𝑙 + 2 and 𝑀 ⊆ [𝑛] over subsets disjoint
with 𝑆. The homogeneous component of degree 𝑑 equals the contributions of the combinations
(𝑆, 𝑀) where |𝑀| = 𝑚 𝑑 − 𝑙 − 1. Since 𝑆 t 𝑀 ⊆ [𝑛] and |𝑆| + |𝑀| = 𝑘 + 𝑑 + 1, it follows that
𝑑 ≤ 𝑛 − 𝑘 − 1.
Since the weight functions on the right-hand side of (9.12) are of degree at most |𝑀| + 𝑘 =
𝑘 + 𝑑 − 𝑙 − 1, the homogeneous component of degree 𝑑 falls inside 𝐻𝑑 . For the other inclusion,
consider 𝜈 = 𝑣 𝑇 for 𝑇 ⊆ [𝑛] with |𝑇 | = 𝑡 𝑘 + 𝑑 + 1. The first equality in (9.15) applies for any
𝑁 ⊆ [𝑛] with |𝑁 | = 𝑘 + 𝑚 + 1 = 𝑡 − 𝑙 − 1. If we pick 𝑁 ⊆ 𝑇, we have that 𝑀 ⊆ 𝑁 ⊆ 𝑇 and we can
write 𝑇 as 𝑇 = 𝑆 t 𝑀 where |𝑆| = |𝑇 | − |𝑀| = 𝑘 + 𝑙 + 2. Thus, each term on the left-most side of
(9.15) is of the form of the boundary expression on the right-hand side of (9.12). By Lemma 9.10
and linearity, it follows that all of 𝐻𝑑 can be realized as homogeneous components of degree 𝑑
of polynomials in Van[RFE𝑙𝑘 ].
For 𝑑 = 𝑙 + 1, up to constant factors, there is only one nontrivial composed boundary
map 𝜕𝑤 [𝑘+1] up to scalar multiplication, namely the map 𝜕𝛼 𝑘 ∧ · · · ∧ 𝛼0 from Corollary 9.8. Thus,
Corollary 9.8 represents the special case of Corollary 9.12 for degree 𝑑 = 𝑙 + 1.
The second equality in (9.15) from Proposition 9.11 leads to another characterization of the
multilinear part of Van[RFE𝑙𝑘 ] in terms of 𝜌 and composed boundary maps, one that is more
technical but to which we can directly apply the generalized image-kernel relationship. This
leads to the following test for membership of multilinear polynomials in Van[RFE𝑙𝑘 ]. Consistent
with the characterization in Corollary 9.12 and with Proposition 5.2, the test decomposes into
independent ones for each of the homogeneous components.
Theorem 9.13. Let 𝑘, 𝑙 ∈ ℕ . For any multilinear polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], 𝑝(RFE𝑙𝑘 ) = 0 if and
only if the homogeneous components 𝑝 (𝑑) of 𝑝 for all degrees 𝑑 satisfy the following requirements:
1. 𝑝 (𝑑) = 0 if 𝑑 ≤ 𝑙 or 𝑑 ≥ 𝑛 − 𝑘.
2. For all 𝑑 = 𝑙 + Δ + 1 with Δ ∈ {0, . . . , 𝑛 − 𝑘 − 𝑙 − 2} and all weight functions 𝑤 1 , . . . , 𝑤 Δ+1 of
degree at most 𝑘 + Δ ,
𝜕𝑤1 ∧ · · · ∧ 𝑤Δ+1 (𝜌−1 (𝑝 (𝑑) )) = 0. (9.17)
Proof. Consider the characterization (9.16) of the homogeneous components 𝐻𝑑 in Corollary 9.12.
We already argued that 𝐻𝑑 only contains the zero polynomial for 𝑑 ≤ 𝑙 and that there are no
terms for 𝑑 ≥ 𝑛 − 𝑘 − 1. This gives us condition 1.
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In the remainder of the proof we consider the requirements for 𝑑 ∈ {𝑙 + 1, . . . , 𝑛 − 𝑘 − 1}.
For multilinear 𝑝, 𝜌−1 (𝑝) is well-defined. Applying 𝜌−1 and the second equality in (9.15), we can
alternately write (9.16) as
𝜌−1 (𝐻𝑑 ) = span 𝜕𝐿𝐵 (Λ 𝑘+𝑑+1 (𝑈)), (9.18)
𝑁
𝐵⊆𝑁
|𝐵|=𝑘+1
where 𝑁 ⊆ [𝑛] can be any fixed subset of size |𝑁 | = 𝑘 + 𝑑 − 𝑙, namely by setting 𝑚 = 𝑑 − 𝑙 − 1,
which we know is non-negative. For easier notation, we pick 𝑁 = [𝑘 + 𝑑 − 𝑙]. By Proposition 9.3
with 𝑤 𝑗 𝐿 𝑁 ,𝑗 and Δ 𝑑 − 𝑙 − 1, we can further rewrite the right-hand side of (9.18) as
Ù
𝜌−1 (𝐻𝑑 ) = ker(𝜕𝐿𝐵 ) ∩ Λ𝑑 (𝑈).
𝑁
𝐵⊆[𝑘+Δ+1]
|𝐵|=Δ+1
Thus 𝑝 (𝑑) ∈ 𝐻𝑑 if and only if
(∀𝐵 ⊆ [𝑘 + Δ + 1] with |𝐵| = Δ + 1) 𝜕𝐿𝐵 (𝜌−1 (𝑝 (𝑑) )) = 0. (9.19)
𝑁
Another application of the second part of Proposition 9.11, this time with 𝑘 ← Δ, 𝑚 ← 𝑘, 𝑡 ← 𝑑,
𝜈 = 𝜌−1 (𝑝 (𝑑) ), and 𝑁 = [𝑘 + Δ + 1], shows that if (9.19) holds for the particular choice of weight
functions 𝑤 𝑗 = 𝐿 𝑁 ,𝑗 , then (9.19) holds for all choices of weight functions 𝑤 𝑗 of degree at most
𝑘 + Δ. The statement follows.
As we will argue in more detail below, by another application of the first part of Proposi-
tion 9.11, it suffices in condition 2 of Theorem 9.13 to consider weight functions of the form
𝑤 𝑗 (𝛼) = 𝜇𝐾 (𝛼)𝛼 Δ−𝑗+1 for 𝑗 ∈ [Δ + 1], where 𝐾 ranges over all subsets of size 𝑘 of some fixed
𝑁 ⊆ [𝑛] with |𝑁 | = 𝑘 + Δ + 1. In this case, (9.17) becomes
𝜕𝜇𝐾 (𝛼)𝛼Δ ∧ · · · ∧ 𝜇𝐾 (𝛼)𝛼0 (𝜌−1 (𝑝 (𝑑) )) = 0. (9.20)
The left-hand side of (9.20) lives in Λ𝑙 (𝑈), and the condition is equivalent to requiring that
the coefficient of 𝑣 𝐿 vanishes for every subset 𝐿 ⊆ [𝑛] \ 𝐾 of size |𝐿| = 𝑙. Those coefficients
can be expressed in terms of evaluations of 𝜕𝐿 𝑝 (𝑑) 𝐾←0 , where we take the partial derivative
with respect to the variables 𝑥 𝑖 for 𝑖 ∈ 𝐿 and set the variables 𝑥 𝑖 for 𝑖 ∈ 𝐾 to zero. Intuitively,
whereas in Lemma 9.10 the effect of the masking factors 𝜇𝑀 was to retain only contributions of
monomials that contain 𝑥 𝑖 for every 𝑖 ∈ 𝑀, in this dual setting the effect of 𝜇𝐾 is to cancel the
contributions of monomials that contain 𝑥 𝑖 for at least one 𝑖 ∈ 𝐾.
Proposition 9.14. Let 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] be a multilinear polynomial, let 𝐾, 𝐿 ⊆ [𝑛] be disjoint subsets
with |𝐾| = 𝑘 and |𝐿| = 𝑙, and Δ ∈ ℕ . Let 𝑐 𝐾,𝐿 denote the coefficient of 𝑣 𝐿 in 𝜕𝜇𝐾 (𝛼)𝛼Δ ∧ · · · ∧ 𝜇𝐾 (𝛼)𝛼0 (𝜌−1 (𝑝)),
and 𝑒 𝐾,𝐿 denote the value of 𝜕𝐿 𝑝| 𝐾←0 upon the substitution 𝑥 𝑖 ← 𝜇𝐾 (𝑎 𝑖 )/𝜇𝐿 (𝑎 𝑖 ) for 𝑖 ∈ [𝑛] \ (𝐾 t 𝐿).
Then 𝑐 𝐾,𝐿 = 𝑒 𝐾,𝐿 /det(𝐴𝐿 ).
Proof. By linearity, it suffices to establish the result for monomials 𝑝 = 𝑥 𝑇 where 𝑇 ⊆ [𝑛]. In
such a case 𝜌−1 (𝑝) = 𝑣 𝑇 /det(𝐴𝑇 ).
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
If 𝐿 * 𝑇, then 𝑐 𝐾,𝐿 vanishes because boundary maps can only remove components from a
wedge product, not insert new components (see (9.5)). On the other hand, 𝜕𝐿 𝑥 𝑇 𝐾←0 is identically
zero because we are taking a partial derivative with respect to a variable that does not appear,
so 𝑒 𝐾,𝐿 vanishes and the equality holds.
If 𝐿 ⊆ 𝑇, then by applying Proposition 9.4 to 𝑣 𝑇 and scaling,
Ö
𝑐 𝐾,𝐿 = (−1)XInv(𝑀,𝐿) 𝜇𝐾 (𝑎 𝑖 ) · det(𝐴 𝑀 )/det(𝐴𝑇 ),
𝑖∈𝑀
where 𝑀 𝑇 \ 𝐿. Note that if 𝐾 ∩ 𝑀 ≠ ∅ then the term 𝑖∈𝑀 𝜇𝐾 (𝑎 𝑖 ) vanishes, hence 𝑐 𝐿 vanishes.
Î
On the other hand, 𝜕𝐿 𝑥 𝑇 𝐾←0 is identically zero because we are setting a variable to zero that
appears in the monomial 𝑥 𝑇 . So, 𝑒 𝐾,𝐿 vanishes and the equality holds.
The remaining cases are those where 𝐿 ⊆ 𝑇 and 𝐾 ∩ 𝑀 = ∅. By (9.14)
©Ö Ö
det(𝐴𝑇 ) = det(𝐴 𝑀t𝐿 ) = (−1)XInv(𝑀,𝐿) · (𝑎 𝑖 − 𝑎 𝑗 )® · det(𝐴 𝑀 ) · det(𝐴𝐿 ).
ª
«𝑖∈𝑀 𝑗∈𝐿 ¬
Combining this with the notation 𝜇𝐿 (𝑎 𝑖 ) 𝑗∈𝐿 (𝑎 𝑖 − 𝑎 𝑗 ), we can rewrite the expression for 𝑐 𝐾,𝐿
Î
as !
Ö 𝜇𝐾 (𝑎 𝑖 ) 1
𝑐 𝐾,𝐿 = · . (9.21)
𝜇𝐿 (𝑎 𝑖 ) det(𝐴𝐿 )
𝑖∈𝑀
On the other hand, we have that 𝜕𝐿 𝑥 𝑇 𝐾←0 = 𝑥 𝑀 , and the value upon the substitution 𝑥 𝑖 ←
𝜇𝐾 (𝑎 𝑖 )/𝜇𝐿 (𝑎 𝑖 ) for 𝑖 ∈ [𝑛] \ (𝐾 t 𝐿) equals
Ö 𝜇𝐾 (𝑎 𝑖 )
𝑒 𝐾,𝐿 = . (9.22)
𝜇𝐿 (𝑎 𝑖 )
𝑖∈𝑀
The result follows by comparing (9.21) and (9.22).
In combination with Theorem 9.13, the connection in Proposition 9.14 yields an alternate
proof of Theorem 1.8. It provides a membership test for the ideal generated by the instantiations
of EVC𝑙𝑘 that, beyond the machinery of alternating algebra developed in this section, only
requires the elementary properties of EVC𝑙𝑘 stated in Proposition 3.4. In particular, it does not
make use of the Zoom Lemma, which we developed as a tool to obviate the need for alternating
algebra after we had obtained our results. Note that the alternate approach to Theorem 1.8 still
relies on the Zoom Lemma for the connection to RFE, namely in the argument that the ideal
generated by the instantiations of EVC𝑙𝑘 includes all of Van[RFE𝑙𝑘 ].
Alternate proof of Theorem 1.8. Consider the membership test given by Theorem 9.13. Condition 1
is equivalent to condition 1 in Theorem 1.8. It remains to argue that condition 2 is equivalent to
condition 2 in Theorem 1.8.
Fix Δ ∈ {0, . . . , 𝑛−𝑘−𝑙−2} and consider Proposition 9.11 with 𝑘 ← Δ, 𝑚 ← 𝑘, 𝑡 ← 𝑑 𝑙+Δ+1,
and 𝜈 = 𝜌−1 (𝑝 (𝑑) ). Set 𝑁 ⊆ [𝑛] to be an arbitrary subset of size 𝑁 = 𝑘 + Δ + 1 and rename the
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I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
set 𝑀 as 𝐾. The application of the first equality in Proposition 9.11 tells us that the combined
requirements (9.17) over all choices of weight functions 𝑤1 , . . . , 𝑤 Δ+1 of degree at most 𝑘 + Δ
are equivalent to the combined requirements (9.20) over all subsets 𝐾 ⊆ 𝑁 of size 𝑘, or, because
of the arbitrariness of 𝑁, over all subsets 𝐾 ⊆ [𝑛] of size 𝑘. The left-hand side of (9.20) is a
linear combination of terms of the form 𝑣 𝐿 , where 𝐿 ⊆ [𝑛] is a subset of size |𝐿| = 𝑑 − Δ − 1 = 𝑙
and is disjoint from 𝐾 because of the masking factor 𝜇𝐾 in all weight functions. Thus, (9.20)
holds if and only if the coefficient 𝑐 𝐾,𝐿,𝑑 of 𝑣 𝐿 on the left-hand side vanishes for every such
𝐿. By Proposition 9.14, 𝑐 𝐾,𝐿,𝑑 = 0 is equivalent to 𝑒 𝐾,𝐿,𝑑 = 0, where 𝑒 𝐾,𝐿,𝑑 denotes the value of
𝜕𝐿 𝑝 (𝑑) 𝐾←0 upon the substitution 𝑥 𝑖 ← 𝜇𝐾 (𝑎 𝑖 )/𝜇𝐿 (𝑎 𝑖 ) for 𝑖 ∈ [𝑛] \ (𝐾 t 𝐿).
In summary, condition 2 in Theorem 9.13 stipulates that for all disjoint subsets 𝐾, 𝐿 ⊆ [𝑛]
with |𝐾| = 𝑘 and |𝐿| = 𝑙,
(∀𝑑 ∈ {𝑙 + 1, . . . , 𝑛 − 𝑘 − 1}) 𝑒 𝐾,𝐿,𝑑 = 0. (9.23)
The value 𝑒 𝐾,𝐿,𝑑 is also the coefficient of degree 𝑑 − 𝑙 of the univariate polynomial in 𝑧 obtained
from 𝜕𝐿 𝑝| 𝐾←0 after the substitution (1.7) from condition 2 in Theorem 1.8. Since the range of 𝑑
in (9.23) covers all terms of this univariate polynomial in 𝑧, (9.23) is equivalent to the polynomial
being zero, which is exactly condition 2 in Theorem 1.8.
Beyond multilinearity. Theorem 9.13 does well for understanding the multilinear elements
of the vanishing ideal. For non-multilinear elements, one may do the following. Let Λ b𝑡 (𝑈) be
𝑡
Λ (𝑈) except that coefficients may be arbitrary polynomials in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ] rather than just
scalars in 𝔽 . The decoder map 𝜌 and boundary maps 𝜕𝑤 carry over to Λ b𝑡 (𝑈) directly, though now
𝜌 is no longer injective. The following variation of Theorem 9.9 characterizes ideal membership
for arbitrary polynomials.
Proposition 9.15. Let 𝑘, 𝑙 ∈ ℕ . For any polynomial 𝑝 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], 𝑝(RFE𝑙𝑘 ) = 0 if and only if
b𝑙+1 (𝑈) with 𝜌(𝜂) = 𝑝 such that, for every weight function 𝑤 of degree at most 𝑘,
there exists 𝜂 ∈ Λ
𝜕𝑤 (𝜂) = 0.
Proof. For the forward direction, we consider polynomials of the form 𝑝 = EVC𝑙𝑘 [𝑆] · 𝑚, where
𝑆 ⊆ [𝑛] with |𝑆| = 𝑘 + 𝑙 + 2 and 𝑚 is a (not necessarily multilinear) monomial in 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ].
One choice of 𝜂 ∈ Λ b𝑙+1 (𝑈) for which 𝜌(𝜂) = 𝑝 is 𝜂 = (−1)(𝑘+1)(𝑙+1) 𝜕𝛼 𝑘 ∧ ∧ 𝛼0 (𝑣 𝑆 )·𝑚, by Lemma 9.7.
···
For this choice of 𝜂 and any weight function 𝑤 of degree at most 𝑘, 𝜕𝑤 (𝜂) = 0. The forward
direction follows since every polynomial 𝑝 for which 𝑝(RFE𝑙𝑘 ) = 0 can be expressed as a linear
combination of polynomials of the described form.
For the backward direction, suppose there exists 𝜂 ∈ Λ b𝑙+1 (𝑈) such that 𝜌(𝜂) = 𝑝 and
𝜕𝑤 (𝜂) = 0 for all weight functions 𝑤 of degree at most 𝑘. We can write 𝜂 = 𝑚 𝜔𝑚 · 𝑚 as a linear
Í
combination of monomials 𝑚 ∈ 𝔽 [𝑥 1 , . . . , 𝑥 𝑛 ], each with coefficient 𝜔𝑚 ∈ Λ𝑙+1 (𝑈). Since 𝜕𝑤
does not affect polynomial coefficients by nonconstant factors, we have that for each 𝑚, 𝜕𝑤 (𝜔𝑚 )
vanishes for all 𝑤 of degree at most 𝑘. Theorem 9.9 implies that 𝜌(𝜕𝑤 (𝜔𝑚 )) is not hit by RFE𝑙𝑘 .
By linearity, 𝜌(𝜂) is not hit by RFE𝑙𝑘 .
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P OLYNOMIAL I DENTITY T ESTING VIA E VALUATION OF R ATIONAL F UNCTIONS
While Proposition 9.15 applies to a broader class of polynomials, it has the drawback
that representing polynomials with Λ b𝑙+1 (𝑈) is too redundant. Specifically, whenever 𝑝 has a
representation in Λ 𝑙+1
b (𝑈), there are many 𝜂 ∈ Λ b𝑙+1 (𝑈) that represent 𝑝, and most of them do
not satisfy the boundary conditions, even when 𝑝 belongs to the vanishing ideal. This erodes
the utility of the characterization. Theorems 9.9 and 9.13 yield straightforward tests: Given 𝑝,
form the unique 𝜂 with 𝜌(𝜂) = 𝑝, and then check whether the boundary conditions hold for 𝜂.
Proposition 9.15, on the other hand, leaves 𝜂 underspecified.
Acknowledgements
We are grateful to Hervé Fournier and Arpita Korwar for their presentation at WACT’18 in Paris
[20]. We are indebted to Gautam Prakriya for helpful discussions and detailed feedback. We
also thank Amir Shpilka and Michael Forbes for comments and encouragement, the anonymous
reviewers for their careful proofreading and interesting suggestions, and the ToC editors for their
thorough work. Finally, we appreciate the partial support for this research by the U.S. National
Science Foundation under Grants No. 1838434, 2137424, and 2312540. Any opinions, findings,
and conclusions or recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of the National Science Foundation.
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AUTHORS
Ivan Hu
Ph. D. student
Department of Computer Science
University of Wisconsin – Madison
Madison, Wisconsin, USA
ilhu wisc edu
hhtps://pages.cs.wisc.edu/~ihu/
Dieter van Melkebeek
Professor
Department of Computer Science
University of Wisconsin – Madison
Madison, Wisconsin, USA
dieter cs wisc edu
https://pages.cs.wisc.edu/~dieter/
T HEORY OF C OMPUTING, Volume 20 (1), 2024, pp. 1–70 69
I VAN H U , D IETER VAN M ELKEBEEK , AND A NDREW M ORGAN
Andrew Morgan
Software engineer
Google
amorgan cs wisc edu
https://pages.cs.wisc.edu/~amorgan/
ABOUT THE AUTHORS
Ivan Hu is a second year Ph. D. student at the University of Wisconsin–Madison
under the supervision of Dieter van Melkebeek. He is studying complexity
theory, with interests in algebraic complexity and pseudorandomness. He is
currently an NSF Graduate Research Fellow.
Dieter van Melkebeek received his Ph. D. from the University of Chicago, under
the supervision of Lance Fortnow. His thesis was awarded the ACM Doctoral
Dissertation Award. After postdocs at DIMACS and the Institute for Advanced
Study, he joined the faculty at the University of Wisconsin-Madison, where
he currently is a full professor. His research interests include the power of
randomness, lower bounds for NP-complete problems, and connections between
derandomization and lower bounds.
Andrew Morgan received his Ph. D. in 2022 from the University of Wiscon-
sin–Madison under the supervision of Dieter van Melkebeek. After graduating,
he became a software engineer at Google.
T HEORY OF C OMPUTING, Volume 20 (1), 2024, pp. 1–70 70