Universal Analytic Gröbner Bases and Tropical Geometry
International audience A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB's for polynomial ideals, leaving open the question...
Published in: | Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation |
---|---|
Main Authors: | , |
Other Authors: | , , |
Format: | Conference Object |
Language: | English |
Published: |
HAL CCSD
2023
|
Subjects: | |
Online Access: | https://hal.science/hal-04386590 https://hal.science/hal-04386590/document https://hal.science/hal-04386590/file/UniversalAnalyticGB.pdf https://doi.org/10.1145/3597066.3597110 |
id |
ftccsdartic:oai:HAL:hal-04386590v1 |
---|---|
record_format |
openpolar |
spelling |
ftccsdartic:oai:HAL:hal-04386590v1 2024-02-11T10:09:11+01:00 Universal Analytic Gröbner Bases and Tropical Geometry Vaccon, Tristan Verron, Thibaut Université de Limoges (UNILIM) Johannes Kepler Universität (JKU) FWF standalone project P34872 Tromsø, Norway 2023-07-24 https://hal.science/hal-04386590 https://hal.science/hal-04386590/document https://hal.science/hal-04386590/file/UniversalAnalyticGB.pdf https://doi.org/10.1145/3597066.3597110 en eng HAL CCSD ACM info:eu-repo/semantics/altIdentifier/arxiv/2401.05759 info:eu-repo/semantics/altIdentifier/doi/10.1145/3597066.3597110 hal-04386590 https://hal.science/hal-04386590 https://hal.science/hal-04386590/document https://hal.science/hal-04386590/file/UniversalAnalyticGB.pdf ARXIV: 2401.05759 doi:10.1145/3597066.3597110 info:eu-repo/semantics/OpenAccess ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023 https://hal.science/hal-04386590 ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromsø, Norway. pp.517-525, ⟨10.1145/3597066.3597110⟩ Algorithms Gröbner bases Tate algebra Tropical Geometry Universal Gröbner basis [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] info:eu-repo/semantics/conferenceObject Conference papers 2023 ftccsdartic https://doi.org/10.1145/3597066.3597110 2024-01-13T23:37:37Z International audience A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB's for polynomial ideals, leaving open the question of how to compute them. In this paper, we provide an algorithm computing a UAGB for a given polynomial ideal, by traversing the Gröbner fan of the ideal. As an application, it offers a new point of view on algorithms for computing tropical varieties of homogeneous polynomial ideals, which typically rely on lifting the computations to an algebra of power series. Motivated by effective computations in tropical analytic geometry, we also examine local bases for more general convergence conditions, constraining the radii to a convex polyhedron. In this setting, we provide an algorithm to compute local Gröbner bases and discuss obstacles towards proving the existence of finite UAGBs. CCS CONCEPTS • Computing methodologies → Algebraic algorithms. Conference Object Tromsø Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Norway Tromsø Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation 517 525 |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
Algorithms Gröbner bases Tate algebra Tropical Geometry Universal Gröbner basis [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] |
spellingShingle |
Algorithms Gröbner bases Tate algebra Tropical Geometry Universal Gröbner basis [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Vaccon, Tristan Verron, Thibaut Universal Analytic Gröbner Bases and Tropical Geometry |
topic_facet |
Algorithms Gröbner bases Tate algebra Tropical Geometry Universal Gröbner basis [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] |
description |
International audience A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB's for polynomial ideals, leaving open the question of how to compute them. In this paper, we provide an algorithm computing a UAGB for a given polynomial ideal, by traversing the Gröbner fan of the ideal. As an application, it offers a new point of view on algorithms for computing tropical varieties of homogeneous polynomial ideals, which typically rely on lifting the computations to an algebra of power series. Motivated by effective computations in tropical analytic geometry, we also examine local bases for more general convergence conditions, constraining the radii to a convex polyhedron. In this setting, we provide an algorithm to compute local Gröbner bases and discuss obstacles towards proving the existence of finite UAGBs. CCS CONCEPTS • Computing methodologies → Algebraic algorithms. |
author2 |
Université de Limoges (UNILIM) Johannes Kepler Universität (JKU) FWF standalone project P34872 |
format |
Conference Object |
author |
Vaccon, Tristan Verron, Thibaut |
author_facet |
Vaccon, Tristan Verron, Thibaut |
author_sort |
Vaccon, Tristan |
title |
Universal Analytic Gröbner Bases and Tropical Geometry |
title_short |
Universal Analytic Gröbner Bases and Tropical Geometry |
title_full |
Universal Analytic Gröbner Bases and Tropical Geometry |
title_fullStr |
Universal Analytic Gröbner Bases and Tropical Geometry |
title_full_unstemmed |
Universal Analytic Gröbner Bases and Tropical Geometry |
title_sort |
universal analytic gröbner bases and tropical geometry |
publisher |
HAL CCSD |
publishDate |
2023 |
url |
https://hal.science/hal-04386590 https://hal.science/hal-04386590/document https://hal.science/hal-04386590/file/UniversalAnalyticGB.pdf https://doi.org/10.1145/3597066.3597110 |
op_coverage |
Tromsø, Norway |
geographic |
Norway Tromsø |
geographic_facet |
Norway Tromsø |
genre |
Tromsø |
genre_facet |
Tromsø |
op_source |
ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023 https://hal.science/hal-04386590 ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromsø, Norway. pp.517-525, ⟨10.1145/3597066.3597110⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/2401.05759 info:eu-repo/semantics/altIdentifier/doi/10.1145/3597066.3597110 hal-04386590 https://hal.science/hal-04386590 https://hal.science/hal-04386590/document https://hal.science/hal-04386590/file/UniversalAnalyticGB.pdf ARXIV: 2401.05759 doi:10.1145/3597066.3597110 |
op_rights |
info:eu-repo/semantics/OpenAccess |
op_doi |
https://doi.org/10.1145/3597066.3597110 |
container_title |
Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation |
container_start_page |
517 |
op_container_end_page |
525 |
_version_ |
1790608970860199936 |