Boundary \({\boldsymbol{H^\ast}}\) -Polynomials of Rational Polytopes.

If is a lattice polytope (i.e., is the convex hull of finitely many integer points in ) of dimension , Ehrhart’s famous theorem [C. R. Acad. Sci. Paris, 254 (1962), pp. 616–618] asserts that the integer-point counting function is a degree- polynomial in the integer variable . Equivalently, the generating function is a rational function of the form ; we call the -polynomial of . There are several known necessary conditions for -polynomials, including results by Hibi [Discrete Math., 83 (1990), pp. 119–121], Stanley [J. Pure Appl. Algebra, 73 (1991), pp. 307–314], and Stapledon [Trans. Amer. Math. Soc., 361 (2009), pp. 5615–5626], who used an interplay of arithmetic (integer-point structure) and topological (local -vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the -polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations.