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| Autors principals: | , , |
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| Format: | Preprint |
| Publicat: |
2024
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2409.14869 |
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| _version_ | 1866916647605895168 |
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| author | Lerario, Antonio Rizzi, Luca Tiberio, Daniele |
| author_facet | Lerario, Antonio Rizzi, Luca Tiberio, Daniele |
| contents | In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative approximate definable choices Lerario, Antonio Rizzi, Luca Tiberio, Daniele Algebraic Geometry Differential Geometry Metric Geometry 14P10, 53C17 In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest. |
| title | Quantitative approximate definable choices |
| topic | Algebraic Geometry Differential Geometry Metric Geometry 14P10, 53C17 |
| url | https://arxiv.org/abs/2409.14869 |