Regularity of Roots of Polynomials

Let Pa(Z) = Z n + ∑n j=1 ajZ n−j be a C curve of monic polynomials, ai ∈ C(I,C) where I ⊂ R is an interval. We show that if k = k(n) is sufficiently large then any choice of continuous roots of Pa is locally absolutely continuous, in a uniform way with respect to maxj ‖aj‖Ck 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.