On the real-rootedness of the veronese construction for rational formal power series

Abstract

We study real sequences {an}nϵN that eventually agree with a polynomial. We prove that if the numerator polynomial of the rational generating series of {an}nϵN is of degree s and has only nonnegative coefficients, then the numerator polynomial of any subsequence {arn+i}nϵN, 0 ≤ i < r, has only nonpositive, real roots for all r ≥ s - i. We show that this bound is optimal. Our results can be applied to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen.Macaulay algebras. In particular, we prove that the Ehrhart h∗-polynomial of the r-th dilate of a d-dimensional polytope has only distinct, negative, real roots if r ≥ min{s + 1,d}. This proves a conjecture of Beck and Stapledon (2010).