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 $${{\mathrm{codim}}}X$$ 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 (Michalek, 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).