A new proof of Bronshtein's theorem

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.