Splittings of Toric Ideals

Favacchio, Giuseppe; Hofscheier, Johannes; Keiper, Graham; Van Tuyl, Adam

Abstract

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.

Citation

Favacchio, G., Hofscheier, J., Keiper, G., & Van Tuyl, A. (2021). Splittings of Toric Ideals. Journal of Algebra, 574, 409-433. https://doi.org/10.1016/j.jalgebra.2021.01.012

• Journal Article Type: Article
• Acceptance Date: Jan 27, 2021
• Online Publication Date: Jan 28, 2021
• Publication Date: May 15, 2021
• Deposit Date: Feb 8, 2021
• Publicly Available Date: Jan 29, 2022
• Journal: Journal of Algebra
• Print ISSN: 0021-8693
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 574
• Pages: 409-433
• DOI: https://doi.org/10.1016/j.jalgebra.2021.01.012
• Keywords: Algebra and Number Theory
• Public URL: https://nottingham-repository.worktribe.com/output/5276260
• Publisher URL: https://www.sciencedirect.com/science/article/abs/pii/S0021869321000375?via%3Dihub