Related papers: Ehrhart polynomial and Successive Minima

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and…

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the…

In algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than…

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just…

In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the…

The multiple root loci among univariate polynomials of degree $n$ are indexed by partitions of $n$. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our…

We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

The n'th Birkhoff polytope is the set of all doubly stochastic n-by-n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these…

In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

This article considers some q-analogues of classical results concerning the Ehrhart polynomials of Gorenstein polytopes, namely properties of their q-Ehrhart polynomial with respect to a good linear form. Another theme is a specific linear…

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of…

Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $\Delta$. We consider lattice polytopes of…

It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax…

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well…

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer…

The strong spectral order induces a natural partial ordering on the manifold $H_{n}$ of monic hyperbolic polynomials of degree $n$. We prove that twisted root maps associated with linear operators acting on $H_{n}$ are G\aa rding convex on…

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair…

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials.…