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Schubert polynomials form a basis of all polynomials and appear in the studyof cohomology rings of flag manifolds. The vanishing problem for Schubertpolynomials asks if a coefficient of a Schubert polynomial is zero. We give atableau criterion to solve this problem, from which we deduce the firstpolynomial time algorithm. These results are obtained from newcharacterizations of the Schubitope, a generalization of the permutahedrondefined for any subset of the n x n grid. In contrast, we show that computingthese coefficients explicitly is #P-complete.

An efficient algorithm for deciding vanishing of Schubert polynomial coefficients