Papers
Topics
Authors
Recent
Search
2000 character limit reached

Computational complexity, Newton polytopes, and Schubert polynomials

Published 24 Oct 2018 in math.CO and cs.CC | (1810.10361v2)

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.

Citations (9)

Summary

No one has generated a summary of this paper yet.

Paper to Video (Beta)

No one has generated a video about this paper yet.

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Continue Learning

We haven't generated follow-up questions for this paper yet.

Collections

Sign up for free to add this paper to one or more collections.