Characteristic polynomials of symmetric matrices over the univariate polynomial ring

JOURNAL OF ALGEBRA

卷:487

Article

Hanselka, Christoph¹ 

[1] Univ Auckland, Dept Math, Private Bag 92019, Auckland 1142, New Zealand

关键词: Characteristic polynomials;    Hyperbolic polynomials;    Determinantal representations;    Different ideal;   

DOI  :  10.1016/j.jalgebra.2017.05.036

来源: Elsevier

【 摘 要 】

Viewing a bivariate polynomial f is an element of R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic polynomial of a symmetric matrix with entries in R[x], it is obviously real rooted. In this article the converse is established, namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring. As a byproduct we present a purely algebraic proof of the Helton-Vinnikov Theorem which solved the 60 year old Lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free