Regularity of roots of polynomials
Résumé
Let $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$ be a $C^k$ curve of monic polynomials, $a_i \in C^k(I,\mathbb C)$ where $I \subset \mathbb R$ is an interval. We show that if $k=k(n)$ is sufficiently large then any choice of continuous roots of $P_a$ is locally absolutely continuous, in a uniform way with respect to $\max_j \|a_j\|_{C^k}$ on compact subintervals. This solves a problem that was open for more then a decade and shows that some systems of pseudodifferential equations are locally solvable. Our main tool is the resolution of singularities.