Regularity of roots of polynomials

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$,$a_j : I \to \mathbb C$with$I \subset \mathbb R$ 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 $a_j\in C^k$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1\le q<p$, in a uniform way with respect to ${\max }_j\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{C^k}$. 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 $P_a$ in terms of its coefficients $a_j$ which we derive using resolution of singularities. For cubic polynomials we compute the formulas as well as bounds for $k$ and $p$ explicitly.