Related papers: Homogeneous multivariate polynomials with the half…
The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…
In this paper, we study a certain type of Hurwitz numbers which count branched covers over the Riemann sphere admitting several branch points with fixed ramification types, one branch point with a fixed number of preimages, and one branch…
In this paper, we exhibit new monotonicity properties of roots of families of orthogonal polynomials $P_n^{(z)}(x)$ depending polynomially on a parameter (Laguerre and Gegenbauer). By establishing that $P_n^{(z)}(x)$ are realrooted in $z$…
We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$…
A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that $h_r(J_1,...,J_n),$ the complete homogeneous…
Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$…
This paper is concerned with the study of $M$-structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal…
This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally…
Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a…
We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type theorems and nonnegative tensors in unified fashion. We prove a weak and strong Perron-Frobenius theorem for these maps and…
Starting from the classical division polynomials we construct homogeneous polynomials $\alpha_n$, $\beta_n$, $\gamma_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP =…
In this paper we study planar hybrid systems composed by two stable linear systems, defined by Hurwitz matrices, in addition with a jump that can be a piecewise linear, a polynomial or an analytic function. We provide an explicit analytic…
Given a set of inequalities determined by homogeneous forms, the following intertwined results are established: (1) the volume of the real semi-algebraic domain determined by these inequalities is explicitly determined; it is shown to be…
A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…
Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Pad\'e polynomials to the given…
We present a novel method for checking the Hurwitz stability of a polytope of matrices. First we prove that the polytope matrix is stable if and only if two homogenous polynomials are positive on a simplex, then through a newly proposed…
Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi\-ma\-troids as a(nother) natural generalization of matroids. As they have shown, demi\-ma\-troids are the appropriate combinatorial objects for studying Wei's duality. Our…