Splittings of Toric Ideals
Giuseppe Favacchio, Johannes Hofscheier, Graham Keiper, Adam Van Tuyl

TL;DR
This paper studies how toric ideals can be decomposed into smaller toric ideals, providing conditions for such splittings and exploring their implications for algebraic invariants like Betti numbers.
Contribution
It introduces a sufficient condition for splitting toric ideals based on their defining matrices and explores specific cases related to graph ideals.
Findings
Provided a criterion for splitting toric ideals using their defining matrices.
Identified additional splittings for graph toric ideals related to subgraphs.
Linked splittings to the computation of Betti numbers of the ideals.
Abstract
Let be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal can be "split" into the sum of two smaller toric ideals. For a general toric ideal , we give a sufficient condition for this splitting in terms of the integer matrix that defines . When is the toric ideal of a finite simple graph , we give additional splittings of related to subgraphs of . When there exists a splitting of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of in terms of the (multi-)graded Betti numbers of and .
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