On the continuity of the solution map for polynomials

TL;DR

This paper investigates the continuity of the roots of monic polynomials as functions of their coefficients, establishing conditions under which the solution map is continuous in various functional spaces, including Sobolev and Wasserstein spaces.

Contribution

It proves the continuity of the roots map for polynomials with coefficients in the $C^d$ class, extending previous results that focused on Sobolev spaces and optimal regularity conditions.

Findings

Solution map is continuous in the $C^d$ coefficient space.

Established bounds linking coefficient regularity to root regularity.

Interpreted results within the Wasserstein space framework.

Abstract

In previous work, we proved that the continuous roots of a monic polynomial of degree $d$ whose coefficients depend in a $C^{d-1,1}$ way on real parameters belong to the Sobolev space $W^{1,q}$ for all $1\le q<d/(d-1)$. This is optimal. We obtained uniform bounds that show that the solution map ``coefficients-to-roots'' is bounded with respect to the $C^{d-1,1}$ and the Sobolev $W^{1,q}$ structures on source and target space, respectively. In this paper, we prove that the solution map is continuous, provided that we consider the $C^d$ structure on the space of coefficients. Since there is no canonical choice of an ordered $d$-tuple of the roots, we work in the space of $d$-valued Sobolev functions equipped with a strong notion of convergence. We also interpret the results in the Wasserstein space on the complex plane.