Ehrhart Theory of Spanning Lattice Polytopes

Abstract

A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^{\ast }$-vector of a spanning lattice polytope has no gaps, i. e., $h_i^{\ast }=0$ implies $h_{i+1}^{\ast }=0$. This generalizes a recent result by Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this relates to unimodality questions of lattice polytopes and previously achieved decomposition results on lattice polytopes of given degree.

17 pages, 7 figures. Minor corrections. To appear in IMRN

References in corpus (2)

• Lattice polytopes having h^*-polynomials with given degree and linear coefficient
• Quotient singularities, integer ratios of factorials and the Riemann Hypothesis

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• The $h^{\ast }$-polynomial of the cut polytope of $K_{2,m}$ in the lattice spanned by its vertices