On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series - Mathematics > CombinatoricsReport as inadecuate




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Abstract: For every positive integer $n$, consider the linear operator $\U {n}$ onpolynomials of degree at most $d$ with integer coefficients defined as follows:if we write $\frac{ht}{1 - t^{d + 1}} = \sum {m \geq 0} gm t^{m}$, forsome polynomial $gm$ with rational coefficients, then $\frac{\U {n}ht}{1-t^{d + 1}} = \sum {m \geq 0} gnm t^{m}$. We show that there exists apositive integer $n {d}$, depending only on $d$, such that if $ht$ is apolynomial of degree at most $d$ with nonnegative integer coefficients and$h0 \geq 1$, then for $n \geq n {d}$, $\U {n}ht$ has simple, real, strictlynegative roots and positive, strictly log concave and strictly unimodalcoefficients. Applications are given to Ehrhart $\delta$-polynomials andunimodular triangulations of dilations of lattice polytopes, as well as Hilbertseries of Veronese subrings of Cohen-MacCauley graded rings.



Author: Matthias Beck, Alan Stapledon

Source: https://arxiv.org/



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