On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

Klara Nosan; Amaury Pouly; Sylvain Schmitz; Mahsa Shirmohammadi; James Worrell

doi:10.1145/3476446.3536172

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

Klara Nosan
, Amaury Pouly
, Sylvain Schmitz
, Mahsa Shirmohammadi
, James Worrell

Laboratoire de Probabilités et Modèles Aléatoires
University of Oxford

Résultats de recherche: Le chapitre dans un livre, un rapport, une anthologie ou une collection › Contribution à une conférence › Revue par des pairs

Résumé

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.

langue originale	Anglais
titre	ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022
rédacteurs en chef	Amir Hashemi
Editeur	Association for Computing Machinery
Pages	129-138
Nombre de pages	10
ISBN (Electronique)	9781450386883
Les DOIs	https://doi.org/10.1145/3476446.3536172
état	Publié - 4 juil. 2022
Modification externe	Oui
Evénement	47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022 - Virtual, Online, France Durée: 4 juil. 2022 → 7 juil. 2022

Série de publications

Nom	Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Une conférence

Une conférence	47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022
Pays/Territoire	France
La ville	Virtual, Online
période	4/07/22 → 7/07/22

Accès au document

10.1145/3476446.3536172

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Contient cette citation

Nosan, K., Pouly, A., Schmitz, S., Shirmohammadi, M., & Worrell, J. (2022). On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices. Dans A. Hashemi (Ed.), ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022 (p. 129-138). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). Association for Computing Machinery. https://doi.org/10.1145/3476446.3536172

Nosan, Klara ; Pouly, Amaury ; Schmitz, Sylvain et al. / On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices. ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022. Editeur / Amir Hashemi. Association for Computing Machinery, 2022. p. 129-138 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

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abstract = "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.",

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Nosan, K, Pouly, A, Schmitz, S, Shirmohammadi, M & Worrell, J 2022, On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices. Dans A Hashemi (Ed.), ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, Association for Computing Machinery, p. 129-138, 47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022, Virtual, Online, France, 4/07/22. https://doi.org/10.1145/3476446.3536172

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices. / Nosan, Klara; Pouly, Amaury; Schmitz, Sylvain et al.
ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022. Ed. / Amir Hashemi. Association for Computing Machinery, 2022. p. 129-138 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

Résultats de recherche: Le chapitre dans un livre, un rapport, une anthologie ou une collection › Contribution à une conférence › Revue par des pairs

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T1 - On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

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AU - Worrell, James

PY - 2022/7/4

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N2 - 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.

AB - 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.

KW - algebraic geometry

KW - algebraic groups

KW - zariski topology

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Nosan K, Pouly A, Schmitz S, Shirmohammadi M, Worrell J. On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices. Dans Hashemi A, Editeur, ISSAC 2022 - Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation47th International Symposium on Symbolic and Algebraic Computation, ISSAC 2022. Association for Computing Machinery. 2022. p. 129-138. (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). doi: 10.1145/3476446.3536172

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

Résumé

Série de publications

Une conférence

Accès au document

Autres fichiers et liens

Empreinte digitale

Contient cette citation