Local description of phylogenetic group-based models

Motivated by phylogenetics, our aim is to obtain a system of 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 phylogenetic invariants (polynomial equations) 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 [Mic12, Conjecture 7.9] and, on the set U , Conjectures 29 and 30 of [SS05].