Markov random fields and iterated toric fibre products

TL;DR

This paper proves that iterated toric fibre products and Markov random fields constructed from finite graphs are defined by binomials of bounded degree, highlighting a uniform bound in their algebraic complexity.

Contribution

It establishes a uniform bound on the degree of binomials defining iterated toric fibre products and associated Markov random fields, advancing understanding of their algebraic structure.

Findings

Bounded degree of binomials for iterated toric fibre products

Uniform Markov degree for Markov random fields from finite graphs

Implication for algebraic complexity of these structures

Abstract

We prove that iterated toric fibre products from a finite collection of toric varieties are defined by binomials of uniformly bounded degree. This implies that Markov random fields built up from a finite collection of finite graphs have uniformly bounded Markov degree.