Hilbert polynomial of the Kimura 3-parameter model

TL;DR

This paper investigates whether the Hilbert polynomial's dependence solely on the number of leaves, as seen in the Jukes Cantor model, extends to the Kimura 3-parameter model, finding that it does not.

Contribution

It demonstrates that unlike the Jukes Cantor model, the Hilbert polynomial for the Kimura 3-parameter model depends on the tree shape.

Findings

Hilbert polynomial varies with tree shape in the Kimura 3-parameter model

Generalization from Jukes Cantor model does not hold for Kimura 3-parameter model

Tree shape influences algebraic properties in the Kimura model

Abstract

Buczy\'{n}ska and Wi\'{s}niewski showed that for the Jukes Cantor binary model of a 3-valent tree the Hilbert polynomial depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. In this paper we consider the Kimura 3-parameter model and show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a 3-valent tree.