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.

Splittings of toric ideals / Favacchio, G.; Hofscheier, J.; Keiper, G.; Van Tuyl, A.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 574:(2021), pp. 409-433. [10.1016/j.jalgebra.2021.01.012]

Splittings of toric ideals

Favacchio G.;
2021

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2872042