Continuity of the solution map for hyperbolic polynomials

TL;DR

This paper proves the continuity of the roots of hyperbolic polynomials with respect to coefficients in certain Sobolev spaces, extending Bronshtein's Lipschitz result and exploring implications for eigenvalues.

Contribution

It establishes the continuity of the solution map from hyperbolic polynomials with $C^d$ coefficients to their roots in Sobolev $W^{1,q}$ spaces, for all finite q, and discusses applications to matrices.

Findings

Continuity of roots in Sobolev $W^{1,q}$ spaces for $1 \,\leq q < \infty$.

Failure of continuity for $q=\infty$.

Applications to eigenvalues of Hermitian matrices.

Abstract

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots with respect to the $C^d$ structure on the source space and the Sobolev $W^{1,q}$ structure, for all $1 \le q<\infty$, on the target space. Continuity fails for $q=\infty$. As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values.