Splittings of Toric Ideals
Abstract
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 $I_2$.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.12820
 Bibcode:
 2019arXiv190912820F
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Combinatorics;
 13D02;
 13P10;
 14M25;
 05E40
 EPrint:
 20 pages, 9 figures