A Note on Nondegenerate Matrix Polynomials

2006

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails.

2008

Abstract. We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature. 1.

arXiv (Cornell University), 2022

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article are families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example,to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

Linear Algebra and its Applications, 2020

We prove that the determinant of a matrix polynomial with Hermitian positive definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.

Following the classical approach of Pólya-Schur theory we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.

Discrete Mathematics & Theoretical Computer Science, 2020

International audience A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Sc...

Filomat, 2011

We show that if ? is a regular semi-classical form (linear functional), then the symmetric form u defined by the relation x?u=??? where ?u is the even part of u, is also regular and semi-classical form for every complex ? except for a discrete set of numbers depending on ?. We give explicitly the recurrence coefficients, integral representation and the structure relation coefficients of the orthogonal polynomials sequence associated with u and the class of the form u knowing that of ?. We conclude with some illustrative examples.

Journal of Fixed Point Theory and Applications, 2008

Following the classical approach of Pólya-Schur theory [14] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials. 2000 Mathematics Subject Classification. Primary: 12D15, 15A04; Secondary: 12D10.

Linear Algebra and its Applications, 1992

Suppose S is any commutative, antinegative semiring (such as the nonegative reals, the nonnegative integers, or a Boolean algebra) and a,, a,, . . . , a, are in §. We show that when n > 1, the operator X + ~~=a=oakXk on the n X n matrices over S is surjective if and only if a, is a unit and all other ak = 0. In a series of papers [l-5], J. V. Brawley and coauthors L. Carliz, R. Gilmer, G. E. Schnibben, and T. Vaughan studied necessary and sufficient conditions for a matrix operator X + a,Z + a,X + . . . + a,Xm induced by a scalar polynomial p(r) = Cy=,,ajxj to be bijective on the n X n matrices over various rings and fields. For example, in [5], Brawley and Schnibben proved that a polynomial with coefficients in an algebraically

Journal of Mathematical Analysis and Applications, 2005

We prove that homogeneous symmetric polynomial inequalities of degree p ∈ {4, 5} in n positive variables can be algorithmically tested, on a finite set depending on the given inequality (Theorem 13); the test-set can be obtained by solving a finite number of equations of degree not exceeding p−2. Section 3 discusses the structure of the ordered vector spaces (ℋ_p^[n],≼) and (ℋ_p^[n],⋞). In Section 4, positivity criteria for degrees 4 and 5 are stated and proved. The main results are Theorems 10–14. Part III of this work will be concerned with the construction of extremal homogeneous symmetric polynomials (best inequalities) of degree 4 in n positive variables.