Splittings of toric ideals

Let I ⊆ R = K [ x 1 , … , 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 .