Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917463187259392 |
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| author | Jones, Abigail R. Lee, Kisun Rodriguez, Jose Israel |
| author_facet | Jones, Abigail R. Lee, Kisun Rodriguez, Jose Israel |
| contents | Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model Jones, Abigail R. Lee, Kisun Rodriguez, Jose Israel Numerical Analysis Computational Complexity Algebraic Geometry 68Q25, 65H10, 65H14 Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation. |
| title | Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model |
| topic | Numerical Analysis Computational Complexity Algebraic Geometry 68Q25, 65H10, 65H14 |
| url | https://arxiv.org/abs/2605.04302 |