Sparse multivariate polynomial interpolation in the basis of Schubert polynomials

TL;DR

This paper develops a deterministic polynomial-time algorithm for sparse interpolation of polynomials in the Schubert basis, leveraging properties of Schubert and skew Schubert polynomials, with applications to computing Littlewood-Richardson coefficients.

Contribution

It introduces a general sparse interpolation algorithm for polynomials in the Schubert basis, extending previous work on symmetric polynomials and derandomizing the process.

Findings

Schubert and skew Schubert polynomials are in CountP and VNP complexity classes.

A polynomial-time deterministic interpolation algorithm for Schubert basis expansions is proposed.

The algorithm enables efficient computation of generalized Littlewood-Richardson coefficients.

Abstract

Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients. In this paper we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in $\CountP$ (when evaluating on non-negative integral inputs) and $\VNP$. Our main result is a deterministic algorithm that computes the expansion of a polynomial $f$ of degree $d$ in $\Z[x_1, \dots, x_n]$ in the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in $n$, $d$, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood-Richardson coefficients.