Polynomials Manipulation Module

Computations with polynomials are at the core of computer algebra and having a fast and robust polynomials manipulation module is a key for building a powerful symbolic manipulation system. SymPy has a dedicated module sympy.polys for computing in polynomial algebras over various coefficient domains.

There is a vast number of methods implemented, ranging from simple tools like polynomial division, to advanced concepts including Gröbner bases and multivariate factorization over algebraic number domains.

Contents

• Basic functionality of the module
  • Introduction
  • Basic functionality
    • Division
    • GCD and LCM
    • Square-free factorization
    • Factorization
    • Groebner bases
    • Solving Equations
• Examples from Wester’s Article
  • Introduction
  • Examples
    • Simple univariate polynomial factorization
    • Univariate GCD, resultant and factorization
    • Multivariate GCD and factorization
    • Support for symbols in exponents
    • Testing if polynomials have common zeros
    • Normalizing simple rational functions
    • Expanding expressions and factoring back
    • Factoring in terms of cyclotomic polynomials
    • Univariate factoring over Gaussian numbers
    • Computing with automatic field extensions
    • Univariate factoring over various domains
    • Factoring polynomials into linear factors
    • Advanced factoring over finite fields
    • Working with expressions as polynomials
    • Computing reduced Gröbner bases
    • Multivariate factoring over algebraic numbers
    • Partial fraction decomposition
  • Literature
• Polynomials Manipulation Module Reference
  • Basic polynomial manipulation functions
  • Extra polynomial manipulation functions
  • Domain constructors
  • Algebraic number fields
  • Monomials encoded as tuples
  • Orderings of monomials
  • Formal manipulation of roots of polynomials
  • Symbolic root-finding algorithms
  • Special polynomials
  • Orthogonal polynomials
  • Manipulation of rational functions
  • Partial fraction decomposition
  • Dispersion of Polynomials
• AGCA - Algebraic Geometry and Commutative Algebra Module
  • Introduction
  • Reference
    • Base Rings
    • Modules, Ideals and their Elementary Properties
    • Module Homomorphisms and Syzygies
• Internals of the Polynomial Manipulation Module
  • Domains
    • Abstract Domains
    • Concrete Domains
    • Implementation Domains
  • Level One
  • Level Zero
    • Manipulation of dense, multivariate polynomials
    • Manipulation of dense, univariate polynomials with finite field coefficients
    • Manipulation of sparse, distributed polynomials and vectors
    • Polynomial factorization algorithms
    • Groebner basis algorithms
  • Exceptions
  • Reference
    • Modular GCD
  • Undocumented
• Series Manipulation using Polynomials
  • rs_series
  • Contribute
    • Manipulation of power series
• Literature