Roots of Garding hyperbolic polynomials

TL;DR

This paper investigates the regularity properties of roots of Garding hyperbolic and real stable polynomials, providing optimal Sobolev space regularity results for eigenvalues and singular values of matrices.

Contribution

It introduces new regularity results for roots of hyperbolic polynomials, extending to eigenvalues and singular values with optimal Sobolev space regularity.

Findings

Regularity results for roots of hyperbolic polynomials

Optimal Sobolev space regularity for eigenvalues

Optimal Sobolev space regularity for singular values

Abstract

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.