Constructive degree bounds for group-based models

TL;DR

This paper establishes degree bounds for ideals defining group-based models in algebraic statistics, proving that for the 3-Kimura model degree 4 suffices, and providing bounds for general G-models.

Contribution

It proves that the 3-Kimura model's defining ideal is generated in degree 4 and establishes a uniform degree bound for G-models across all trees.

Findings

The 3-Kimura model's ideal is generated in degree 4.

Existence of a uniform degree bound for G-models.

Degree bounds facilitate testing membership in the model variety.

Abstract

Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant. We prove that for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant $d$, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.