Computational complexity, Newton polytopes, and Schubert polynomials
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 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.
Cite
@article{arxiv.1810.10361,
title = {Computational complexity, Newton polytopes, and Schubert polynomials},
author = {Anshul Adve and Colleen Robichaux and Alexander Yong},
journal= {arXiv preprint arXiv:1810.10361},
year = {2021}
}
Comments
12 pages. v2 is the conference proceedings version. The remaining material of v1 (proofs, the theorem on #P-completeness) is split off into a subsequent arXiv post. That post has a different title ("An efficient algorithm for deciding vanishing of Schubert polynomial coefficients"), abstract, and discussion of the general "nonvanishing problem" removed