Se p 20 03 CHOOSING ROOTS OF POLYNOMIALS SMOOTHLY , II

We show that the roots of any smooth curve of polynomials with real roots only can be parametrized twice differentiable (but not better). In [1] we claimed that there exists a smooth curve of polynomials of degree 3 for which no C-parametrization of the roots exists. Unfortunately there was an error in the calculation of b3 and we have been informed by Jacques Chaumat and Anne-Marie Chollet in June 2001 about that and the related papers [2], [5]. We are now going to repair this mistake and improve at the same time the results of [2]. The smoothness assumptions in the following theorem are certainly not the best possible but in fact we are mainly interested in the case of smooth coefficients. The conclusion of the theorem is the best possible, since even for the characteristic polynomial of a smooth curve of symmetric matrices there needn’t be a differentiable parametrization of the roots with locally Hölderian derivative as the first example in [3] shows. Let P be a curve defined on some subset T ⊆ R of monic polynomials P (t) of degree n ≥ 1 with real roots only. A parametrization of some class of the roots of P is a curve x : T → R of that class such that for each t ∈ T the values x1(t), . . . , xn(t) are the roots of P (t) with correct multiplicity. Theorem. Consider a continuous curve of polynomials P (t)(x) = x − a1(t)x n−1 + · · ·+ (−1)an(t), t ∈ R, with all roots real. Then there is a continuous parametrization x = (x1, . . . , xn) : R → R of the roots of P . Moreover: (1) [2], Theorem 1 and Theorem 2. If all coefficients ai are of class C n then the parametrization x : R → R may be chosen differentiable with locally