Inequalities and Ehrhart $δ$-Vectors

Details Inequalities and Ehrhart $δ$-Vectors Inequalities and Ehrhart $δ$-Vectors

2008-01-06

arxiv.org/abs/0801.0873v2

Trans. Amer. Math. Soc. 361 (2009), 5615-5626.

Mathematics

Combinatorics

11 pages. v2: minor changes, more detailed proof of Lemma 2.12. To appear in Trans. Amer. Math. Soc

Scientific paper

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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