On the Log-concavity of Hilbert Series of Veronese Subrings and Ehrhart Series

For every positive integer n, consider the linear operator Un on polynomials of degree at most d with integer coefficients defined as follows: if we write h(t) (1−t)d+1 = P m≥0 g(m) t , for some polynomial g(m) with rational coefficients, then Un h(t) (1−t)d+1 = P m≥0 g(nm) t . We show that there exists a positive integer nd, depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) ≥ 1, then for n ≥ nd, Un h(t) has simple, real, strictly negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–MacCauley graded rings.