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Computing algebraic degrees of phylogenetic varieties

Luis David García Puente, Marina Garrote-López and Elima Shehu

Vol. 14 (2023), No. 2, 215–231

Phylogenetic varieties are algebraic varieties specified by a statistical model describing the evolution of biological sequences along a tree. Its understanding is an important problem in algebraic statistics, particularly in the context of phylogeny reconstruction. In the broader area of algebra statistics, there have been important theoretical advances in computing certain invariants associated with algebraic varieties arising in applications. Beyond the dimension and degree of a variety, one is interested in computing other algebraic degrees, such as the maximum likelihood degree and the Euclidean distance degree. Despite these efforts, the current literature lacks explicit computations of these invariants for the particular case of phylogenetic varieties. In our work, we fill this gap by computing these invariants for phylogenetic varieties arising from the simplest group-based models of nucleotide substitution Cavender–Farris–Neyman model, Jukes–Cantor model, Kimura 2-parameter model and the Kimura 3-parameter model on small phylogenetic trees with at most 5 leaves.

phylogenetics, euclidean distance degree, maximum likelihood degree, numerical algebraic geometry, symbolic computations
Mathematical Subject Classification
Primary: 62R01, 65H10, 92D15
Secondary: 14Q30
Received: 30 September 2022
Revised: 18 January 2024
Accepted: 6 February 2024
Published: 16 May 2024
Luis David García Puente
Department of Mathematics and Computer Science
Colorado College
Colorado Springs, CO
United States
Marina Garrote-López
Max Planck Institute for Mathematics in the Sciences
Elima Shehu
Osnabrück University