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 \(|nP ∩ Z^d|\) is a degree- \(d\) polynomial in the integer variable \(n\) . Equivalently, the generating function \(1 + ∑n ≥ 1 |nP ∩ Z^d| \, z^n\) 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 \(h\) -vectors 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.Keywordslattice polytoperational polytopeEhrhart quasipolynomialboundary \(h^*\) -polynomialsymmetric decompositionGorenstein polytopeMSC codes52B2052C0705A15