Local equations for equivariant evolutionary models

Marta Casanellas, Jesus Fernandez-Sanchez*, Mateusz Michalek

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

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", see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states,K, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied ("d-claw tree hypothesis"), we provide an algorithm to produce a complete intersection that describes the variety around these points. (C) 2017, Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)285-323
Number of pages39
JournalADVANCES IN MATHEMATICS
Volume315
DOIs
Publication statusPublished - 31 Jul 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Evolutionary model
  • Complete intersection
  • Phylogenetic variety
  • Phylogenetic tree
  • Representation theory
  • GENERAL MARKOV MODEL
  • PHYLOGENETIC INVARIANTS
  • LINEAR INVARIANTS
  • SECANT VARIETIES
  • TREES
  • GEOMETRY
  • IDEALS
  • IDENTIFIABILITY
  • LIKELIHOOD
  • SEQUENCES

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