Multigraded Commutative Algebra of Graph Decompositions

Abstract

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to $K_4$-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.

33 pages, v2:typos corrected, v3: Section 5 merged into Section 3 with Conj 5.1 now Thm. 3.6, v4: shortened and improved presentation, v5: final published version

References in corpus (3)

• Sequential importance sampling for multiway tables
• Algebraic varieties representing group-based Markov processes on trees
• Decompositions of Binomial Ideals in Macaulay 2

Cited by in corpus (8)

• Positive margins and primary decomposition
• Maximum Likelihood Estimation of Toric Fano Varieties
• Toric ideals of series and parallel connections of matroids
• Toric fiber products versus Segre products
• Rational partition models under iterative proportional scaling
• The geometry of Gaussian double Markovian distributions
• Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing
• The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models