BOUNDARY H*-POLYNOMIALS OF RATIONAL POLYTOPES

If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R-d) of dimension d, Ehrhart's famous theorem [C. R. Acad. Sci. Paris, 254 (1962), pp. 616-618] asserts that the integer-point counting function |nP boolean AND Z(d)| is a degree-d polynomial in the integer variable n. Equivalently, the generating function 1 + Sigma(n >= 1) | nP boolean AND Z(d)| z(n) is a rational function of the form h*(z)/(1-z)(d+1); we call h*(z) the h*-polynomial of P. There are several known necessary conditions for h*-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 hvectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the h*-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.