Natural reparameterization and continuity of the solution map for polynomials

The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is $W^{1,q}$, for $q \in [1,\frac{d}{d-1})$, where $d$ is the degree. This result is stable in the sense that the solution map from $C^d$ coefficients to $W^{1,q}$ roots is continuous. We prove that, after a n