A new proof of Bronshtein's theorem

TL;DR

This paper presents a new, self-contained proof of Bronshtein's theorem, establishing the Lipschitz continuity of continuous roots of hyperbolic polynomial families and providing explicit bounds for the Lipschitz constant.

Contribution

It offers a novel proof of Bronshtein's theorem and derives explicit bounds for the Lipschitz constant of roots, also revalidating recent results on smoothness of roots.

Findings

Proof of Lipschitz continuity of roots for $C^{n-1,1}$-families

Explicit bounds for the Lipschitz constant of roots

Reproof of $C^n$-curve root smoothness result

Abstract

We give a new self-contained proof of Bronshtein's theorem, that any continuous root of a $C^{n-1,1}$-family of monic hyperbolic polynomials of degree $n$ is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product we reprove the recent result of Colombini, Orr\'u, and Pernazza, that a $C^n$-curve of hyperbolic polynomials of degree $n$ admits a $C^1$-system of its roots.