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Interlacing polynomials and the Veronese construction for rational formal power series. (English) Zbl 1437.05018

Summary: Fixing a positive integer \(r\) and \(0\leq k\leq r-1\), define \(f^{\langle r,k\rangle}\) for every formal power series \(f\) as \(f(x)=f^{\langle r,0\rangle}(x^r)+xf^{\langle r,1\rangle}(x^r)+\cdots+x^{r-1}f^{\langle r,r-1 \rangle}(x^r)\). K. Jochemko [Int. Math. Res. Not. 2018, No. 15, 4780–4798 (2018; Zbl 1408.13045)] 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 \(r\geqslant\deg h(x)-k\) and any positive integer \(n\). As a consequence, K. Jochemko [loc. cit.] confirmed a conjecture of M. Beck and A. Stapledon [Math. Z. 264, No. 1, 195–207 (2010; Zbl 1230.05017)] 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 \(r\geqslant n\). In this paper, we provide an alternative approach to Beck and Stapledon’s conjecture by proving the following general result: if the polynomial sequence \(( h^{\langle r,r-i\rangle}(x))_{1\leq i\leq r}\) is interlacing, so is \((U^n_{r,r-i}h(x))_{1\leq i\leq r}\). Our result has many other interesting applications. In particular, this enables us to give a new proof of C. D. Savage and M. Visontai’s result [Trans. Am. Math. Soc. 367, No. 2, 1441–1466 (2015; Zbl 1316.05002)] 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.

MSC:

05A15 Exact enumeration problems, generating functions
13A02 Graded rings
13C14 Cohen-Macaulay modules
26C10 Real polynomials: location of zeros
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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References:

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