Polynomial Manipulation

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 concepts
    • Polynomials
    • Divisibility
  • 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
  • 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
    • Finite Extensions
• Introducing the Domains of the poly module
  • What are the domains?
  • Representing expressions symbolically
    • Tree representation
    • DUP representation
    • DMP representation
    • Sparse polynomial representation
  • Basic usage of domains
  • Domain elements vs sympy expressions
  • Gaussian integers and Gaussian rationals
  • Finite fields
  • Real and complex fields
  • Algebraic number fields
  • Polynomial ring domains
  • Old (dense) polynomial rings
  • PolyRing vs PolynomialRing
  • Rational function fields
  • Expression domain
  • Choosing a domain
  • Converting elements between different domains
  • Unifying domains
  • Internals of a Poly
  • Choosing a domain for a Poly
  • Choosing generators
  • Algebraically dependent generators
• Reference docs for the Poly Domains
  • Domains
  • Abstract Domains
  • GF(p)
  • ZZ
  • QQ
  • MPQ
  • Gaussian domains
  • ZZ_I
  • QQ_I
  • QQ<a>
  • RR
  • CC
  • K[x]
  • K(x)
  • EX
  • Quotient ring
  • Sparse polynomials
  • Sparse rational functions
  • Dense polynomials
• Internals of the Polynomial Manipulation Module
  • 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
  • Options
  • Configuration
  • Exceptions
  • Reference
    • Modular GCD
  • Undocumented
• Series Manipulation using Polynomials
  • rs_series
  • Contribute
    • Manipulation of power series
• Literature
• Poly solvers
  • solve_lin_sys
  • eqs_to_matrix
  • sympy_eqs_to_ring
  • _solve_lin_sys
  • _solve_lin_sys_component
• Introducing the domainmatrix of the poly module
  • What is domainmatrix?
  • DomainMatrix Class Reference
  • DDM Class Reference
  • SDM Class Reference
  • Normal Forms
• Number Fields
  • Introduction
  • Solving the Main Problems
    • Integral Basis
    • Prime Decomposition
    • p-adic Valuation
    • Finding Minimal Polynomials
    • The Subfield Problem
  • Internals
    • Algebraic number fields
    • Representing algebraic numbers
    • Finitely-generated modules
    • Utilities

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Basic functionality of the module