A new proof of Bronshtein’s theorem

Abstract

We give a new self-contained proof of Bronshtein’s theorem, that any continuous root of a [Formula: see text]-family of monic hyperbolic polynomials of degree [Formula: see text] is locally Lipschitz continuous, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product, we re-prove the recent result of Colombini, Orrú, and Pernazza that a [Formula: see text]-curve of hyperbolic polynomials of degree [Formula: see text] admits a [Formula: see text]-system of its roots.