Low degree equations for phylogenetic group-based models

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 , we provide an explicit construction of polynomial equations (phylogenetic invariants) of degree at most that define the variety on a Zariski open set . The set contains all biologically meaningful points when is the group of the Kimura 3-parameter model. In particular, our main result confirms (Michaek, Toric varieties: phylogenetics and derived categories, PhD thesis, Conjecture 7.9, 2012) and, on the set , Conjectures 29 and 30 of Sturmfels and Sullivant (J Comput Biol 12:204-228, 2005).