Some Lower Bounds on the Reach of an Algebraic Variety
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913369708036096 |
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| author | La Valle, Chris Tonelli-Cueto, Josué |
| author_facet | La Valle, Chris Tonelli-Cueto, Josué |
| contents | Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's $γ$ and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15649 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Lower Bounds on the Reach of an Algebraic Variety La Valle, Chris Tonelli-Cueto, Josué Algebraic Geometry Computational Geometry Symbolic Computation 14Q20, 14P99, 14Q65 Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's $γ$ and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions. |
| title | Some Lower Bounds on the Reach of an Algebraic Variety |
| topic | Algebraic Geometry Computational Geometry Symbolic Computation 14Q20, 14P99, 14Q65 |
| url | https://arxiv.org/abs/2402.15649 |