Computability of Equivariant Gröbner bases
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918090249338880 |
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| author | Ghosh, Arka Lopez, Aliaume |
| author_facet | Ghosh, Arka Lopez, Aliaume |
| contents | Let $\mathbb{K}$ be a field, $\mathcal{X}$ be an infinite set (of indeterminates), and $\mathcal{G}$ be a group acting on $\mathcal{X}$. An ideal in the polynomial ring $\mathbb{K}[\mathcal{X}]$ is called equivariant if it is invariant under the action of $\mathcal{G}$. We show Gröbner bases for equivariant ideals are computable are hence the equivariant ideal membership is decidable when $\mathcal{G}$ and $\mathcal{X}$ satisfies the Hilbert's basis property, that is, when every equivariant ideal in $\mathbb{K}[\mathcal{X}]$ is finitely generated. Moreover, we give a sufficient condition for the undecidability of the equivariant ideal membership problem. This condition is satisfied by the most common examples not satisfying the Hilbert's basis property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computability of Equivariant Gröbner bases Ghosh, Arka Lopez, Aliaume Logic in Computer Science Commutative Algebra Let $\mathbb{K}$ be a field, $\mathcal{X}$ be an infinite set (of indeterminates), and $\mathcal{G}$ be a group acting on $\mathcal{X}$. An ideal in the polynomial ring $\mathbb{K}[\mathcal{X}]$ is called equivariant if it is invariant under the action of $\mathcal{G}$. We show Gröbner bases for equivariant ideals are computable are hence the equivariant ideal membership is decidable when $\mathcal{G}$ and $\mathcal{X}$ satisfies the Hilbert's basis property, that is, when every equivariant ideal in $\mathbb{K}[\mathcal{X}]$ is finitely generated. Moreover, we give a sufficient condition for the undecidability of the equivariant ideal membership problem. This condition is satisfied by the most common examples not satisfying the Hilbert's basis property. |
| title | Computability of Equivariant Gröbner bases |
| topic | Logic in Computer Science Commutative Algebra |
| url | https://arxiv.org/abs/2507.08990 |