Complexity, combinatorial positivity, and Newton polytopes
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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∩ coNP of problems with "good characterizations". This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on 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.
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