Abstract
For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write \({\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}\) , for some polynomial g(m) with rational coefficients, then \({\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}}\) . We show that there exists a positive integer n d , 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 ≥ n d , U n h(t) has simple, real, 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–Macauley graded rings.
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The authors would like to thank Alexander Barvinok, Jesus De Loera, Sergey Fomin, Joseph Gubeladze, Mircea Mustaţǎ, Sam Payne, John Stembridge, Volkmar Welker, and an anonymous referee for useful discussions and helpful suggestions. M. Beck was partially supported by the NSF (research grant DMS-0810105), and A. Stapledon was partially supported by an Eleanor Sophia Wood travelling scholarship from the University of Sydney.
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Beck, M., Stapledon, A. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series. Math. Z. 264, 195–207 (2010). https://doi.org/10.1007/s00209-008-0458-7
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DOI: https://doi.org/10.1007/s00209-008-0458-7