Multivariate Polynomials in Sage

TL;DR

The paper presents a Sage software patch that implements multivariate polynomials with various bases, enabling advanced algebraic computations and basis definitions using divided difference operators.

Contribution

It introduces a new Sage patch for multivariate polynomials with multiple bases, including the ability to define and expand polynomials in these bases.

Findings

Implementation includes Schubert, Key, Grothendieck, and Macdonald polynomials.

Supports double-variable sets and specific double-linear bases.

Allows defining new bases via divided difference operators.

Abstract

We have developed a patch implementing multivariate polynomials seen as a multi-base algebra. The patch is to be released into the software Sage and can already be found within the Sage-Combinat distribution. One can use our patch to define a polynomial in a set of indexed variables and expand it into a linear basis of the multivariate polynomials. So far, we have the Schubert polynomials, the Key polynomials of types A, B, C, or D, the Grothendieck polynomials and the non-symmetric Macdonald polynomials. One can also use a double set of variables and work with specific double-linear bases like the double Schubert polynomials or double Grothendieck polynomials. Our implementation is based on a definition of the basis using divided difference operators and one can also define new bases using these operators.