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arXiv · 2511.22778

Recent Advances in the Theory of Polyomino Ideals

Francesco Navarra·Ayesha Asloob Qureshi
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Abstract

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-Poincar\'e 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.

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Francesco Navarra, Ayesha Asloob Qureshi. 2025-11-27. Recent Advances in the Theory of Polyomino Ideals. https://doi.org/10.65908/gja.2025.24978

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