Splittings of Toric Ideals

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 $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ 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 $I_2$.