The three-state toric homogeneous Markov chain model has Markov degree two

TL;DR

This paper proves that the three-state toric homogeneous Markov chain model's associated toric ideals are generated by quadratic binomials, confirming a conjecture and simplifying the algebraic structure of these models.

Contribution

It establishes that the Markov degree for the three-state toric homogeneous Markov chain model is two, showing the ideals are generated by quadratic binomials, which was previously conjectured.

Findings

The Markov degree of the model is two.

Toric ideals are generated by quadratic binomials.

Confirms the conjecture by Haws et al.

Abstract

We prove that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals are generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by binomials of degree six or less.