Year: 2026
Type: preprint
Abstract: 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 natural Lipschitz reparameterization, the roots are Lipschitz and the map from $C^d$ coefficients to reparameterization and reparameterized roots is continuous with respect to the $W^{1,q}$ topology on the target spaces, for all $q \in [1,\infty)$. The result is based on a convergence and reparameterization theorem for Almgren's $Q$-valued Sobolev functions.
Source: arXiv (Cornell University)
Authors: Parusiński, Adam, Rainer, Armin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
OpenAlex ID: W7215042034