Generalized Binomial Edge Ideals

TL;DR

This paper introduces a generalized class of binomial edge ideals linked to graphs, exploring their algebraic properties, Gr"obner bases, and primary decompositions, with implications for conditional independence models.

Contribution

It extends binomial edge ideals to a broader class, providing methods to compute Gr"obner bases and primary decompositions using combinatorial graph properties.

Findings

Gr"obner bases are square-free and can be computed via graph paths.

Generalized binomial edge ideals are radical.

Irreducible components are all rational.

Abstract

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by studying paths in the graph. Since these Gr\"obner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.