We show that smooth curves of monic complex polynomials Pa(Z)=Zn+∑j=1najZn−j, aj:I→C with I⊂R a compact interval, have absolutely continuous roots in a uniform
way. More precisely, there exists a positive integer k and a rational number
p>1, both depending only on the degree n, such that if aj∈Ck
then any continuous choice of roots of Pa is absolutely continuous with
derivatives in Lq for all 1≤q<p, in a uniform way with respect to
maxj∥aj∥Ck. The uniformity allows us to deduce also a multiparameter
version of this result. The proof is based on formulas for the roots of the
universal polynomial Pa in terms of its coefficients aj which we derive
using resolution of singularities. For cubic polynomials we compute the
formulas as well as bounds for k and p explicitly.Comment: 32 pages, 2 figures; minor changes; accepted for publication in Ann.
Sc. Norm. Super. Pisa Cl. Sci. (5); some typos correcte