Splittings of Toric Ideals

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

Density of ff-ideals and ff-ideals in mixed small degrees

A squarefree monomial ideal is called an $f$-ideal if its Stanley–Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees