Related papers: Nonnegative Polynomials and Circuit Polynomials

We develop a new kind of nonnegativity certificate for univariate polynomials on an interval. In many applications, nonnegative Bernstein coefficients are often used as a simple way of certifying polynomial nonnegativity. Our proposed…

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…

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x+1), which leads to the log-concavity of…

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not…

We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible…

We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These…

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

We show that Thom polynomials of Lagrangian singularities have nonnegative coefficients in the basis consisting of Q-functions. The main tool in the proof is nonnegativity of cone classes for globally generated bundles.

The classical version of P\'olya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

Let $n_1 < n_2 < \cdots < n_N$ be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials $T_N(\theta) = \sum_{j=1}^N {\cos (n_j\theta)}$ tends to $\infty$ as a…

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…

In this article, we are interested in the enumeration of Fully Packed Loops configurations on a grid with a given noncrossing matching. These quantities also appear as the groundstate components of the Completely Packed Loops model as…

A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct…

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the $k$-element subsets of $[n]$. For simplicity, we…

This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums…

A polynomial system with $n$ equations in $n$ variables supported on a set $\mathcal{W}\subset\mathbb{R}^n$ of $n+2$ points has at most $n+1$ non-degenerate positive solutions. Moreover, if this bound is reached, then $\mathcal{W}$ is…

Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…