Sparse multivariate polynomial interpolation on the basis of Schubert polynomials

摘要

Schubert polynomials were discovered by A. Lascoux and M. Schutzenberger in the study of cohomology rings of flag manifolds in 1980s. 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 #P (when evaluating on nonnegative integral inputs) and VNP. Our main result is a deterministic algorithm that computes the expansion of a polynomial f of degree d in on 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 (Adv Appl Math 18(3):271-285, 1997). 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.