Low degree equations for phylogenetic group-based models

Abstract

Motivated by phylogenetics, our aim is to obtain a system of low degree equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group G, we provide an explicit construction of codimX polynomial equations (phylogenetic invariants) of degree at most |G| that define the variety X on a Zariski open set U. The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result confirms (Michałek, Toric varieties: phylogenetics and derived categories, PhD thesis, Conjecture 7.9, 2012) and, on the set U, Conjectures 29 and 30 of Sturmfels and Sullivant (J Comput Biol 12:204–228, 2005).