Numerical Nonlinear Algebra
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917606295863296 |
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| author | Bates, Daniel J. Breiding, Paul Chen, Tianran Hauenstein, Jonathan D. Leykin, Anton Sottile, Frank |
| author_facet | Bates, Daniel J. Breiding, Paul Chen, Tianran Hauenstein, Jonathan D. Leykin, Anton Sottile, Frank |
| contents | Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations.
Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08585 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical Nonlinear Algebra Bates, Daniel J. Breiding, Paul Chen, Tianran Hauenstein, Jonathan D. Leykin, Anton Sottile, Frank Algebraic Geometry Numerical Analysis 14Q65, 65H14 Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications. |
| title | Numerical Nonlinear Algebra |
| topic | Algebraic Geometry Numerical Analysis 14Q65, 65H14 |
| url | https://arxiv.org/abs/2302.08585 |