On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866912257203503104 |
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| author | Nosan, Klara Pouly, Amaury Schmitz, Sylvain Shirmohammadi, Mahsa Worrell, James |
| author_facet | Nosan, Klara Pouly, Amaury Schmitz, Sylvain Shirmohammadi, Mahsa Worrell, James |
| contents | We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups, where all the groups are generated over a fixed number field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_01853 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices Nosan, Klara Pouly, Amaury Schmitz, Sylvain Shirmohammadi, Mahsa Worrell, James Computational Complexity Algebraic Geometry Group Theory We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups, where all the groups are generated over a fixed number field. |
| title | On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices |
| topic | Computational Complexity Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2106.01853 |