Comprehensive Systems for Primary Decompositions of Parametric Ideals
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929478027968512 |
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| author | Ishihara, Yuki Yokoyama, Kazuhiro |
| author_facet | Ishihara, Yuki Yokoyama, Kazuhiro |
| contents | We present an effective method for computing parametric primary decomposition via comprehensive Gröbner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients $\mathbb{Q}[A,X]$ where $A$ is the set of parameters and $X$ is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs $(C, \mathcal{Q})$, where for each parameter value in $C$, the ideal has the stable decomposition $\mathcal{Q}$. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set $\mathcal{O}$ such that $φ_α(\mathcal{Q})$ is a primary decomposition for any $α\in C\cap \mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comprehensive Systems for Primary Decompositions of Parametric Ideals Ishihara, Yuki Yokoyama, Kazuhiro Symbolic Computation Commutative Algebra We present an effective method for computing parametric primary decomposition via comprehensive Gröbner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients $\mathbb{Q}[A,X]$ where $A$ is the set of parameters and $X$ is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs $(C, \mathcal{Q})$, where for each parameter value in $C$, the ideal has the stable decomposition $\mathcal{Q}$. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set $\mathcal{O}$ such that $φ_α(\mathcal{Q})$ is a primary decomposition for any $α\in C\cap \mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition. |
| title | Comprehensive Systems for Primary Decompositions of Parametric Ideals |
| topic | Symbolic Computation Commutative Algebra |
| url | https://arxiv.org/abs/2408.15917 |