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Register a new userInterlacing Ehrhart Polynomials of Reflexive Polytopes
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It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.
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In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,ldots$ has the interlacing property.
Let $P(b)subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=sum_{y}int_{P(b)cap (y+L)} h(x) mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ such that the shape of $P(b)$ does not change when $b$ runs over a chamber. We first prove that on every chamber of $R^N$, $S^L (P(b),h)$ is given by a quasi-polynomial function of $bin R^N$. A key point of our paper is an analysis of the interplay between two notions of degree on quasi-polynomials: the usual polynomial degree and a filtration, called the local degree. Then, for a fixed $kleq d$, we consider a particular linear combination of such intermediate weighted sums, which was introduced by Barvinok in order to compute efficiently the $k+1$ highest coefficients of the Ehrhart quasi-polynomial which gives the number of points of a dilated rational polytope. Thus, for each chamber, we obtain a quasi-polynomial function of $b$, which we call Barvinoks patched quasi-polynomial (at codimension level $k$). Finally, for each chamber, we introduce a new quasi-polynomial function of $b$, the cone-by-cone patched quasi-polynomial (at codimension level $k$), defined in a refined way by linear combinations of intermediate generating functions for the cones at vertices of $P(b)$. We prove that both patched quasi-polynomials agree with the discrete weighted sum $bmapsto S^0(P(b),h)$ in the terms corresponding to the $k+1$ highest polynomial degrees.
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivaria
A graph whose nodes have degree 1 or 3 is called a ${1,3}$-graph. Liu and Osserman associated a polytope to each ${1,3}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set ${0,frac14,frac12,1}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.
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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