Multivariate Polynomials in Sage

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.