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 $C^{n-1,1}$-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, Orrú, and Pernazza, that a $C^n$-curve of hyperbolic polynomials of degree $n$ admits a $C^1$-system of its roots.

15 pages; explicit bounds for the Lipschitz constant of the root added; proof that the roots can be chosen $C^1$ if the coefficients are $C^n$ added; small corrections and numbering changed; accepted for publication in J. Hyperbolic Differ. Equ; some typos corrected

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