On the Coefficients of Hilbert Quasipolynomials

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert–Serre). We derive an upper bound for its grade, i. e. the index from which on its coefficients are constant. As an application, we give a purely algebraic proof of an old combinatorial result (due to Ehrhart, McMullen and Stanley). 1. HILBERT QUASIPOLYNOMIALS Let K be a field, and R a positively graded K-algebra, that is, R = ⊕ i≥0 Ri where R0 = K and R is finitely generated over K. We do not assume R to be generated in degree 1 – the generators may be of arbitrarily high degree. The theorem of Hilbert–Serre describes the Hilbert functions of finitely generated graded R-modules M: Theorem 1. Let M = ⊕ i∈Z Mi be a finitely generated graded R-module of dimension d, H(M, ) : Z → Z the associated Hilbert function, and suppose that r1, . . . ,rd is a homogeneous system of parameters for M. Then there is a quasi-polynomial QM of degree d −1, such that H(M,n) = QM(n) for n ≫ 0. Moreover, the period of QM divides lcm(degr1, . . . ,rd). The terminology concerning quasipolynomials is explained as follows: a function Q : Z → C is called a quasi-polynomial of degree u if Q(n) = au(n)n u +au−1(n)n u−1 + . . .+a1(n)n+a0(n), where ai : Z → C is a periodic function for i = 0, . . . ,u, and au 6= 0. The period of Q is the smallest positive integer π such that ai(n+mπ) = ai(n) for all n,m ∈ Z and i = 0, . . . ,u. For the reader’s convenience, we include a short proof the Hilbert–Serre theorem, or rather its reduction to the classical theorem of Hilbert. By definition of homogeneous system of parameters, M is a finitely generated module over K[r1, . . . ,rd] (which is isomorphic to a polynomial ring over K). Therefore we may assume that R = K[r1, . . . ,rd]. Let S be the subalgebra of R generated by its homogeneous elements of degree p = lcm(degr1, . . . ,degrd). Then it is not hard to see that R is a finitely generated S-module. Therefore M is a finitely generated S-module, too, and dimS M = dimR M. As a last reduction step, we can replace R by S and assume that R is generated by its elements of degree p. Then we have the decomposition M = M0 ⊕ . . .⊕Mp−1, Mk = ⊕