Radical splittings of toric ideals

Let K be a field and let IA⊂ K[x₁,… ,xₙ] I A ⊂ K [ x 1 , … , x n ] be a toric ideal. We study when IA I A can be expressed in the form IA=rad(IA₁+⋯ +IAᵣ), I A = rad ( I A 1 + ⋯ + I A r ) , where IAᵢ≠ IA I A i ≠ I A for every i . In particular, we provide a necessary and sufficient condition for such a decomposition with $$r=2$$ r = 2 . We also introduce the radical splitting number of IA I A , denoted by Splitrad(IA) Split rad ( I A ) , and compute its exact value for several classes of toric ideals, with particular emphasis on toric ideals arising from graphs. Specifically, we show that Splitrad(IA)=3 Split rad ( I A ) = 3 for toric ideals of complete bipartite graphs, except for the toric ideal of K2,2 K 2 , 2 . We also prove that Splitrad(IA) Split rad