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On the real-rootedness of the Veronese construction for rational formal power series by Jochemko, Katharina is a scholarly article available to read on EtoBox.

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We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. We apply our results 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 $h^\ast$-polynomial of the $r$-th dilate of a $d$-dimensional polytope has only distinct, negative, real roots if $r\geq \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).

Author

Jochemko, Katharina

Published

2016

Language

EN