Radical splittings of toric ideals
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912889273581568 |
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| author | Katsabekis, Anargyros Thoma, Apostolos |
| author_facet | Katsabekis, Anargyros Thoma, Apostolos |
| contents | Let $I_A \subset K[x_1,\ldots,x_n]$ be a toric ideal. In this paper, we provide a necessary and sufficient condition for the toric variety $V(I_A)$, over an algebraically closed field, to be expressed as the set-theoretic intersection of other toric varieties. We also introduce the radical splitting number of $I_A$, denoted by ${\rm Split}_{\rm rad}(I_A)$, and compute its exact value in several cases, with particular emphasis on toric ideals arising from graphs. In particular, we show that ${\rm Split}_{\rm rad}(I_A)=3$ for toric ideals of complete bipartite graphs. Additionally, we prove that ${\rm Split}_{\rm rad}(I_A)$ coincides with the binomial arithmetical rank of $I_A$ when the height of $I_A$ is equal to 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Radical splittings of toric ideals Katsabekis, Anargyros Thoma, Apostolos Commutative Algebra Algebraic Geometry 13F65, 05C25, 05E40, 14M25 Let $I_A \subset K[x_1,\ldots,x_n]$ be a toric ideal. In this paper, we provide a necessary and sufficient condition for the toric variety $V(I_A)$, over an algebraically closed field, to be expressed as the set-theoretic intersection of other toric varieties. We also introduce the radical splitting number of $I_A$, denoted by ${\rm Split}_{\rm rad}(I_A)$, and compute its exact value in several cases, with particular emphasis on toric ideals arising from graphs. In particular, we show that ${\rm Split}_{\rm rad}(I_A)=3$ for toric ideals of complete bipartite graphs. Additionally, we prove that ${\rm Split}_{\rm rad}(I_A)$ coincides with the binomial arithmetical rank of $I_A$ when the height of $I_A$ is equal to 2. |
| title | Radical splittings of toric ideals |
| topic | Commutative Algebra Algebraic Geometry 13F65, 05C25, 05E40, 14M25 |
| url | https://arxiv.org/abs/2507.15785 |