A tropical method for solving parametrized polynomial systems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912171091296256 |
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| author | Helminck, Paul Alexander Henriksson, Oskar Ren, Yue |
| author_facet | Helminck, Paul Alexander Henriksson, Oskar Ren, Yue |
| contents | We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A tropical method for solving parametrized polynomial systems Helminck, Paul Alexander Henriksson, Oskar Ren, Yue Algebraic Geometry 14T90, 65H14, 13P15 We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs. |
| title | A tropical method for solving parametrized polynomial systems |
| topic | Algebraic Geometry 14T90, 65H14, 13P15 |
| url | https://arxiv.org/abs/2409.13288 |