Radical splittings of toric ideals

Source abstract

Abstract Let K be a field and let $$I_A\subset K[x_1,\ldots ,x_n]$$ I A ⊂ K [ x 1 , … , x n ] be a toric ideal. We study when $$I_A$$ I A can be expressed in the form $$I_A=\textrm{rad}(I_{A_1}+\cdots +I_{A_r}),$$ I A = rad ( I A 1 + ⋯ + I A r ) , where $$I_{A_i}\ne I_A$$ 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 $$I_A$$ I A , denoted by $$\textrm{Split}_{\textrm{rad}}(I_A)$$ 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 $$\textrm{Split}_{\textrm{rad}}(I_A)=3$$ Split rad ( I A ) = 3 for toric ideals of complete bipartite graphs, except for the toric ideal of $$K_{2,2}$$ K 2 , 2 . We also prove that $$\textrm{Split}_{\textrm{rad}}(I_A)$$ Split rad ( I A ) coincides with the binomial arithmetical rank for toric ideals of height 2.