Type of Publication:  Journal article 
Publication status:  Published 
Author:  Casanellas, Marta; FernándezSánchez, Jesús; Michalek, Mateusz 
Year of publication:  2017 
Published in:  Advances in Mathematics ; 315 (2017).  pp. 285323.  Elsevier.  ISSN 00018708.  eISSN 10902082 
DOI (citable link):  https://dx.doi.org/10.1016/j.aim.2017.05.003 
Summary: 
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 κ, 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 (“dclaw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.

Subject (DDC):  510 Mathematics 
Keywords:  Evolutionary model, Complete intersection, Phylogenetic variety, Phylogenetic tree, Representation theory 
Refereed:  Yes 
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CASANELLAS, Marta, Jesús FERNÁNDEZSÁNCHEZ, Mateusz MICHALEK, 2017. Local equations for equivariant evolutionary models. In: Advances in Mathematics. Elsevier. 315, pp. 285323. ISSN 00018708. eISSN 10902082. Available under: doi: 10.1016/j.aim.2017.05.003
@article{Casanellas2017Local52317, title={Local equations for equivariant evolutionary models}, year={2017}, doi={10.1016/j.aim.2017.05.003}, volume={315}, issn={00018708}, journal={Advances in Mathematics}, pages={285323}, author={Casanellas, Marta and FernándezSánchez, Jesús and Michalek, Mateusz} }
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