Optimal Sobolev regularity of roots of polynomials

TL;DR

This paper investigates the optimal regularity of roots of parameter-dependent polynomials, establishing precise integrability conditions for their derivatives and demonstrating the limits of smoothness based on coefficient regularity.

Contribution

It provides the optimal Sobolev regularity results for roots of polynomials with smooth coefficients and generalizes Glaeser inequalities, with applications to pseudo-differential equations and orbit space mappings.

Findings

Roots of $C^{n-1,1}$ polynomial families are locally absolutely continuous with $p$-integrable derivatives for $p < n/(n-1)$.

The regularity results are optimal; roots can have unbounded variation if coefficients are less smooth.

Applications include solvability of pseudo-differential systems and Sobolev regularity of multi-valued functions.

Abstract

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.