Recent Advances in the Theory of Polyomino Ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908818864078848 |
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| author | Navarra, Francesco Qureshi, Ayesha Asloob |
| author_facet | Navarra, Francesco Qureshi, Ayesha Asloob |
| contents | Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a unified account of recent advances on two central themes: the characterization of prime polyomino ideals and the emerging connection between the Hilbert-Poincaré series and Gorensteinness of $K[\mathcal{P}]$ with the classical rook theory. Some further related properties, as radicality, primary decomposition, and levelness are discussed, and a \textit{Macaulay2} package, namely \texttt{PolyominoIdeals}, is also presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_22778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recent Advances in the Theory of Polyomino Ideals Navarra, Francesco Qureshi, Ayesha Asloob Commutative Algebra Combinatorics 05A15, 05B50, 05E40, 68W30 Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a unified account of recent advances on two central themes: the characterization of prime polyomino ideals and the emerging connection between the Hilbert-Poincaré series and Gorensteinness of $K[\mathcal{P}]$ with the classical rook theory. Some further related properties, as radicality, primary decomposition, and levelness are discussed, and a \textit{Macaulay2} package, namely \texttt{PolyominoIdeals}, is also presented. |
| title | Recent Advances in the Theory of Polyomino Ideals |
| topic | Commutative Algebra Combinatorics 05A15, 05B50, 05E40, 68W30 |
| url | https://arxiv.org/abs/2511.22778 |