0000000001043169
AUTHOR
Adam Van Tuyl
Regularity and h-polynomials of toric ideals of graphs
For all integers 4 ≤ r ≤ d 4 \leq r \leq d , we show that there exists a finite simple graph G = G r , d G= G_{r,d} with toric ideal I G ⊂ R I_G \subset R such that R / I G R/I_G has (Castelnuovo–Mumford) regularity r r and h h -polynomial of degree d d . To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.
Splittings of Toric Ideals
Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ 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 $I_2$.