A new iterative algorithm for comprehensive Grobner systems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910417089986560 |
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| author | Bigatti, Anna Maria Palezzato, Elisa Torielli, Michele |
| author_facet | Bigatti, Anna Maria Palezzato, Elisa Torielli, Michele |
| contents | A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K.
The recent algorithms for computing them comprehensive Grobner systems consider the corresponding ideal J in K[A,X], and are based on stability of Grobner bases of ideals under specializations of the parameters. Starting from a Grobner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some ``coefficients'' in K[A].
In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Grobner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using ``comparatively-cheap'' ideal-membership tests, instead of radical-membership tests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new iterative algorithm for comprehensive Grobner systems Bigatti, Anna Maria Palezzato, Elisa Torielli, Michele Commutative Algebra A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them comprehensive Grobner systems consider the corresponding ideal J in K[A,X], and are based on stability of Grobner bases of ideals under specializations of the parameters. Starting from a Grobner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some ``coefficients'' in K[A]. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Grobner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using ``comparatively-cheap'' ideal-membership tests, instead of radical-membership tests. |
| title | A new iterative algorithm for comprehensive Grobner systems |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2404.13514 |