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A new proof of Bronshtein’s theorem

We give a new self-contained proof of Bronshtein’s theorem, that any continuous root of a Cn−1,1-family of monic hyperbolic polynomials of degree n 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, Orru, and Pernazza that a Cn-curve of hyperbolic polynomials of degree n admits a C1-system of its roots.