On the regularity of the roots of hyperbolic polynomials

We prove that a hyperbolic monic polynomial whose coefficients are functions of class C ^r of a parameter t admits roots of class C ¹ in t, if r is the maximal multiplicity of the roots as t varies. Moreover, if the coefficients are functions of t of class C ^2r , then the roots may be chosen two times differentiable at every point in t. This improves, among others, previous results of Bronšteĭn, Mandai, Wakabayashi and Kriegl, Losik and Michor.