PreprintCC BY 4.0arXiv
Natural reparameterization and continuity of the solution map for polynomials
Abstract
The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is W1,q, for q ∈ [1,d/(d−1)), where d is the degree. This result is stable in the sense that the solution map from Cd coefficients to W1,q roots is continuous. We prove that, after a natural Lipschitz reparameterization, the roots are Lipschitz and the map from Cd coefficients to reparameterization and reparameterized roots is continuous with respect to the W1,q topology on the target spaces, for all q ∈ [1,∞). The result is based on a convergence and reparameterization theorem for Almgren's Q-valued Sobolev functions.
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BibTeX
@misc{parusinski2026natural,
title = {{Natural reparameterization and continuity of the solution map for polynomials}},
author = {Adam Parusiński and Armin Rainer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39389},
eprint = {2609.39389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39389},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Natural reparameterization and continuity of the solution map for polynomials
AU - Adam Parusiński
AU - Armin Rainer
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39389
UR - https://arxiv.org/abs/2609.39389
AB - The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is W^1,q, for q ∈ [1,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 ∈ [1,∞). The result is based on a convergence and reparameterization theorem for Almgren's Q-valued Sobolev functions.
ER -