On the continuity of the solution map for polynomials
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912055398760448 |
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| author | Parusiński, Adam Rainer, Armin |
| author_facet | Parusiński, Adam Rainer, Armin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the continuity of the solution map for polynomials Parusiński, Adam Rainer, Armin Functional Analysis Algebraic Geometry Classical Analysis and ODEs Differential Geometry Metric Geometry 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. |
| title | On the continuity of the solution map for polynomials |
| topic | Functional Analysis Algebraic Geometry Classical Analysis and ODEs Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2410.01326 |