Related papers: Preserving positive polynomials and beyond

It is known that the Ehrhart polynomials of cross-polytopes, as well as of pyramids over them, have positive coefficients. We give a combinatorial proof of this fact by showing that a scaled version of the Ehrhart polynomials are generating…

In this paper we give a short proof of the $\ell^p$-improving property of the average operator along the square integers and more general quadratic polynomials. Moreover we obtain a similar result for some higher degree polynomials. We also…

In this paper, using doubly stochastic operators, we have extended the notion of majorization to the space $\ell^p(I)$, where $I$ is assumed to be an infinite set, and then, in the case $p\in (1,+\infty)$, characterize the structure of all…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

Let $\mathbb{F}$ be a finite field and let $f$ be a linear polynomial in $\mathbb{F}[x]$. We investigate the number of polynomials of degree $d$ which commute with $f$ under composition. In so doing, we rediscover a result of Park, but with…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…

We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to…

Given a differential operator of geometric origin there exists a list of operations that preserve this property, e.g., tensor products, pull-backs, push-forwards and the middle convolution. We apply certain sequences of these operations to…

We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of F p (x). Our methods rely on specific tools arising in positive characteristic: p-curvature, structure of simple…

In this paper, we propose a class of explicit positivity preserving numerical methods for general stochastic differential equations which have positive solutions. Namely, all the numerical solutions are positive. Under some reasonable…

In 2009, Borcea and Br\"and\'en characterize all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic…

We will remark an extension of a linear functional on subalgebra of algebra of continuous functions on subset of $\mathbb{R}^n$ which preserves positivity.

We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open…

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the…

Suppose one has a party of $m$ people, whose expertise collectively covers $n$ topics. Given a subset $T$ of the topics, one wishes to form a panel of $|T|$ people from the party such that $T$ can be covered by assigning a distinct topic to…

We examine a result of A. Iserles and E. B. Saff, use it to prove a conjecture of S. Fisk that a linear operator which maps monomials to Legendre polynomials also preserves zeros in the open interval $|x|<1$, and state a more general…

Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…

Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…