Related papers: Preserving positive polynomials and beyond

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by…

In this article we discuss the solvability of some class of fully nonlinear equations, and equations with p-Laplacian in more general conditions by using a new approach given in [1] for studying the nonlinear continuous operator. Moreover…

Linear operators preserving the direct sum of polynomial rings P(m)\oplus P(n) are constructed. In the case |m-n|=1 they correspond to atypical representations of the superalgebra osp(2,2). For |m-n|=2 the generic, finite dimensional…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

P\'olya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the…

As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators…

In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…

Let ${\bf M}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over an arbitrary field $\mathbb{F}$. We consider linear maps $\Phi: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ preserving matrices annihilated by a fixed…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

Thw purpose of this paper is to present a systemic study of some families of the generalized q-Euler numbers and polynomials of higher order.

We study linear functions on the space of $n \times n$ matrices over a field which preserve or strongly preserve each of Green's equivalence relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) and the corresponding…

For every closed subset $X$ of a stratifiable [resp. metrizable] space $Y$ we construct a positive linear extension operator $T:R^{X\times X}\to R^{Y\times Y}$ preserving constant functions, bounded functions, continuous functions,…

In this paper we establish some applications of the Scherer-Hol's theorem for polynomial matrices. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a…

For linear operators which factor with suitable assumptions concerning commutativity of the factors, we introduce several notions of a decomposition. When any of these hold then questions of null space and range are subordinated to the same…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

We investigate linear operators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$. We give explicit operators $A$ such that, for fixed $d\in\mathbb{N}_0$ and closed $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq…

Replacing the usual notion of quotient sets by the notion of orbiquotient sets we obtain a generalization of P\'olya theory. The key ingredient of our extended theory is the definition of the orbicycle index polynomial which we compute in…

In this paper, using the method proposed by Dembo and Mukherjee [5], we obtain the persistence exponents of random Weyl polynomials in both cases: half nonnegative axis and the whole real axis. Our result is a confirmation to the…