Constructive degree bounds for group-based models

Mateusz Michalek*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant (2005) [25, Conjectures 29, 30]. We prove that for the 3-Kimura model, corresponding to the group G = Z(2) X Z(2), the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant d, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d. (C) 2013 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)1672-1694
Number of pages23
JournalJournal of Combinatorial Theory Series A
Volume120
Issue number7
DOIs
Publication statusPublished - Sep 2013
MoE publication typeA1 Journal article-refereed

Keywords

  • Group-based model
  • 3-Kimura
  • Phylogenetic invariant
  • Toric variety
  • PHYLOGENETIC INVARIANTS
  • NUCLEOTIDE-SEQUENCES
  • TORIC IDEALS
  • VARIETIES
  • TREES
  • FINITENESS
  • GEOMETRY

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