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Register a new userInterlacing Polynomials and the Veronese Construction for Rational Formal Power Series
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Authors
Philip B. Zhang
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Fixing a positive integer $r$ and $0 le k le r-1$, 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 $U^{n}_{r,k}, h(x) := left( (1+x+cdots+x^{r-1})^{n} h(x) right)^{langle r,k rangle}$ has only nonpositive zeros for any $r ge deg h(x) -k$ 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 $rge n$. In this paper, we provide an alternative approach to Beck and Stapledons conjecture by proving the following general result: if the polynomial sequence $left( h^{langle r,r-i rangle}(x)right)_{1le i le r}$ is interlacing, so is $left( U^{n}_{r,r-i}, h(x) right)_{1le i le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontais result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations.
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