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Interlacing polynomials and the veronese construction for rational formal power series

Part of: General commutative ring theory Polytopes and polyhedra Polynomials, rational functions Combinatorics

Published online by Cambridge University Press:  30 January 2019

Philip B. Zhang
Philip B. Zhang*
Affiliation:
College of Mathematical Science, Tianjin Normal University, Tianjin, 300387, P. R. China ([email protected])
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Abstract

Fixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension n, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ges n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$ is interlacing, so is $( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.

Keywords

Veronese submoduleEhrhart polynomialsinterlacing polynomialscoloured permutationCarlitz identity

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions 13A02: Graded rings 26C10: Polynomials
Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 1 - 16
Copyright
Copyright © Royal Society of Edinburgh 2019

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