Continuity of the solution map for hyperbolic polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911414693658624 |
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| author | Parusiński, Adam Rainer, Armin |
| author_facet | Parusiński, Adam Rainer, Armin |
| contents | Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots with respect to the $C^d$ structure on the source space and the Sobolev $W^{1,q}$ structure, for all $1 \le q<\infty$, on the target space. Continuity fails for $q=\infty$. As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity of the solution map for hyperbolic polynomials Parusiński, Adam Rainer, Armin Functional Analysis Algebraic Geometry Classical Analysis and ODEs Differential Geometry Metric Geometry Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots with respect to the $C^d$ structure on the source space and the Sobolev $W^{1,q}$ structure, for all $1 \le q<\infty$, on the target space. Continuity fails for $q=\infty$. As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values. |
| title | Continuity of the solution map for hyperbolic polynomials |
| topic | Functional Analysis Algebraic Geometry Classical Analysis and ODEs Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2410.01321 |