Multigraded Commutative Algebra of Graph Decompositions

TL;DR

This paper explores the algebraic structure of graph decompositions through toric fiber products, providing new methods for generating ideals, analyzing properties like normality, and applying these to models in algebraic statistics.

Contribution

It introduces techniques for generating sets of toric fiber products in non-zero codimension and studies their algebraic properties, with applications to statistical models and graph theory.

Findings

Constructed Markov bases for hierarchical models in new cases

Provided a new proof for quartic generation of binary graph models

Developed recursive methods for primary decomposition of ideals

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.