We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{ n-1,1}$-curve of monic polynomials of degree $n$ is locally absolutely continuous with locally $p$-integrable derivatives for every $1 \le p < n/(n-1)$, uniformly with respect to the coefficients. This result is optimal: in general, the derivatives of the roots of a smooth curve of monic polynomials of degree $n$ are not locally $n/(n-1)$-integrable, and the roots may have locally unbounded variation if the coefficients are only of class $C^{ n-1,\alpha}$ for $\alpha <1$. We also prove a generalization of Ghisi and Gobbino's higher order Glaeser inequalities. We give three applications of the main results: local solvability of a system of pseudo-differential equations, a lifting theorem for mappings into orbit spaces of finite group representations, and a sufficient condition for multi-valued functions to be of Sobolev class $W^{ 1,p}$ in the sense of Almgren.
Categories: math.CA