Splittings of toric ideals

Abstract

Let I⊆R=K[x1,…,xn] 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=IG is the toric ideal of a finite simple graph G, we give additional splittings of IG related to subgraphs of G. When there exists a splitting I=I1+I2 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 I1 and I2.