Session A2 - Computational Algebraic Geometry
July 11, 15:00 ~ 15:25 - Room B5
Local equations for equivariant evolutionary models
Universitat Politècnica de Catalunya, Spain - firstname.lastname@example.org
Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open "salmon conjecture") and it is not clear how to use all generators in practice. However, for phylogenetic reconstruction purposes, the elements of the ideal that could be useful only need to describe the variety around certain points of no evolution. With this idea in mind, we produce a collection of explicit equations that describe the variety on a neighborhood of these points. Namely, for any tree on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states , we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution, and provide an algorithm to produce a complete intersection that describes the variety around these points.
Joint work with Marta Casanellas (Universitat Politècnica de Catalunya) and Mateusz Michalek (Polish Academy of Sciences).