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arXiv · 1909.12820

Splittings of Toric Ideals

Giuseppe Favacchio·Johannes Hofscheier·Graham Keiper·Adam Van Tuyl
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Abstract

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.

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Giuseppe Favacchio, Johannes Hofscheier, Graham Keiper, Adam Van Tuyl. 2019-09-27. Splittings of Toric Ideals. https://doi.org/10.1016/j.jalgebra.2021.01.012

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