Inequalities and Ehrhart 𝛿-vectors

Stapledon, A.

doi:10.1090/S0002-9947-09-04776-X

Inequalities and Ehrhart $\delta$-vectors
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Trans. Amer. Math. Soc. 361 (2009), 5615-5626 Request permission

Abstract:

For any lattice polytope $P$, we consider an associated polynomial $\bar {\delta }_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope.

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