OPTIMAL REGULARITY OF ROOTS OF POLYNOMIALS

Overview & Synopsis

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 ≤ 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,α for α < 1. We also give a generalization of Ghisi and Gobbino's higher order Glaeser inequalities.