Multivariate Polynomials and Polynomial Rings

Sage implements multivariate polynomial rings through several backends. The most generic implementation uses the classes sage.rings.polynomial.polydict.PolyDict and sage.rings.polynomial.polydict.ETuple to construct a dictionary with exponent tuples as keys and coefficients as values.

Additionally, specialized and optimized implementations over many specific coefficient rings are implemented via a shared library interface to SINGULAR; and polynomials in the boolean polynomial ring

\[\GF{2}[x_1,...,x_n]/ \langle x_1^2+x_1,...,x_n^2+x_n \rangle.\]

are implemented using the PolyBoRi library (cf. sage.rings.polynomial.pbori).

• Term orders
• Base class for multivariate polynomial rings
• Base class for elements of multivariate polynomial rings
• Multivariate Polynomial Rings over Generic Rings
• Generic Multivariate Polynomials
• Ideals in multivariate polynomial rings
• Polynomial Sequences
• Multivariate Polynomials via libSINGULAR
• Direct low-level access to SINGULAR’s Groebner basis engine via libSINGULAR
• Solution of polynomial systems using msolve
• PolyDict engine for generic multivariate polynomial rings
• Compute Hilbert series of monomial ideals
• Class to flatten polynomial rings over polynomial ring
• Monomials