Computational complexity, Newton polytopes, and Schubert polynomials

TL;DR

This paper introduces a polynomial-time algorithm for determining the nonvanishing of Schubert polynomial coefficients by leveraging combinatorial criteria and geometric structures like the Schubitope, connecting algebraic combinatorics with complexity theory.

Contribution

It provides the first polynomial-time nonvanishing test for Schubert polynomials using new combinatorial and geometric characterizations.

Findings

Nonvanishing problem for Schubert polynomials is in P.

Introduction of the Schubitope as a key geometric object.

A tableau criterion for nonvanishing of Schubert polynomial coefficients.

Abstract

The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class $NP\cap coNP$ of problems with "good characterizations". This suggests a new algebraic combinatorics viewpoint on complexity theory. This report discusses the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid, together with a theorem of A. Fink, K. M\'{e}sz\'{a}ros, and A. St. Dizier, which proved a conjecture of C. Monical, N. Tokcan, and the third author.