Related papers: Real-rooted polynomials and a generalised Hermite-…

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…

Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance…

Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the…

June Huh and Matthew Stevens conjectured that the Hilbert-Poincar\'e series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities…

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ by the use of matrix…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to…

Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Br\"and\'en and Solus have given sufficient conditions under which the image…

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries…

A real polynomial $f$ is called local nonnegative at a point $p$, if it is nonnegative in a neighbourhood of $p$. In this paper, a sufficient condition for determining this property is constructed. Newton's principal part of $f$ (denoted as…

We study in detail the class of even polynomials and their behavior with respect to finite free convolutions. To this end, we use some specific hypergeometric polynomials and a variation of the rectangular finite free convolution to…

This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given…

We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof…

We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational…

A polynomial $p \in \mathbb{R}[z_1, \cdots, z_n]$ is called real stable if it is non-vanishing whenever all the variables take values in the upper half plane. A well known result of Elliott Lieb and Alan Sokal states that if $p$ and $q$ are…