Thin Polytopes: Lattice Polytopes With Vanishing Local h*-Polynomial

Abstract

Abstract In this paper, we study the novel notion of thin polytopes: lattice polytopes whose local $h^{*}$-polynomials vanish. The local $h^{*}$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov, and Mavlyutov; Schepers; and Katz and Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov, and Zelevinsky, where in this case the local $h^{*}$-polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local $h^{*}$-polynomial with a survey of previous results.