On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series
By Matthias Beck, A. Stapledon. 2008
53 citations recorded by the source corpus.
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 $${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}$$ , 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 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, Unh(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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/583935af7618beb3131fccb87e7c4304255076ba
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In Algebra and Number Theory
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Citations in the corpus
- Unimodality Questions for Integrally Closed Lattice Polytopes (2011, result): “We remark that this notion also appears in recent work of Beck and Stapledon [4].”
- Unimodality problems in Ehrhart theory (2015, unknown): “Theorem 17 (Beck and Stapledon, [11]).”