Related papers: On Matrix Polynomials with Real Roots

The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values…

This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…

We prove that a finite graded simplicial poset with a top element added has real-rooted Chow and augmented Chow polynomials whenever it has a positive $h$-vector. This class of posets include Cohen-Macaulay simplicial posets and in…

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect…

Zarhin showed that a matrix constructed from a polynomial with distinct roots has co-rank one. Some striking properties of this matrix are used to give a direct proof of his result. An account is given of calculations carried out to try to…

Over a decade ago De Loera, Haws and K\"oppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding $h^*$-polynomials form a unimodal sequence. The first of…

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…

It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization…

An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is…

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is a semiring with the two operations: addition $a \oplus b := \max(a,b)$ and multiplication $a \otimes b := a + b$. Roots of the characteristic polynomial of a max-plus matrix are called…

We present a proof of the Casas-Alvero conjecture, stating that if a complex polynomial has a root in common with each of its derivatives it must be a multiple of the power of some monomial.

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…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the equivalence of orthosymmetry and orthogonal additivity.

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory…

The polynomial eigenvalue problem arises in many applications and has received a great deal of attention over the last decade. The use of root-finding methods to solve the polynomial eigenvalue problem dates back to the work of…