Related papers: Ehrhart positivity for marked order polytopes

We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…

We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar and by Putinar. We explain how these results can be understood as…

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…

We seek simple conditions on a pair of labeled posets that determine when the difference of their $(P,\omega)$-partition enumerators is $F$-positive, i.e., positive in Gessel's fundamental basis. This is a quasisymmetric analogue of the…

The aim of this paper is to show that if an order preserving bijective transformation of the Hilbert space effect algebra also preserves the probability with respect to a fixed pair of mixed states, then it is an ortho-order automorphism. A…

In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group $\mathcal{S}_{N}$. The…

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial…

Monotone triangles are a rich extension of permutations that biject with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous…

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We…

We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and…

We describe an efficient method for computing the Ehrhart polynomial of Gelfand--Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart--Macdonald reciprocity: evaluating the Ehrhart polynomial at negative…

In Ehrhart theory, the $h^*$-vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic…

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to…

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n…

We discover that tautological intersection numbers on $\bar{\mathcal{M}}_{g, n}$, the moduli space of stable genus $g$ curves with $n$ marked points, are evaluations of Ehrhart polynomials of partial polytopal complexes. In order to prove…

We show that any symmetric positive definite homogeneous matrix polynomial $M\in\R[x_1,...,x_n]^{m\times m}$ admits a piecewise semi-certificate, i.e. a collection of identites $M(x)=\sum_jf_{i,j}(x)U_{i,j}(x)^TU_{i,j}(x)$ where…