dorsal/arxiv
View SchemaToric ideals of phylogenetic invariants
| Authors | Bernd Sturmfels, Seth Sullivant |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0402015 |
| URL | https://arxiv.org/abs/q-bio/0402015 |
Abstract
Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gr\"obner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr\"obner bases consist of certain explicitly constructed polynomials of degree at most four.
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"abstract": "Statistical models of evolution are algebraic varieties in the space of joint\nprobability distributions on the leaf colorations of a phylogenetic tree. The\nphylogenetic invariants of a model are the polynomials which vanish on the\nvariety. Several widely used models for biological sequences have transition\nmatrices that can be diagonalized by means of the Fourier transform of an\nabelian group. Their phylogenetic invariants form a toric ideal in the Fourier\ncoordinates. We determine generators and Gr\\\"obner bases for these toric\nideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr\\\"obner\nbases consist of certain explicitly constructed polynomials of degree at most\nfour.",
"arxiv_id": "q-bio/0402015",
"authors": [
"Bernd Sturmfels",
"Seth Sullivant"
],
"categories": [
"q-bio.PE",
"math.AC"
],
"title": "Toric ideals of phylogenetic invariants",
"url": "https://arxiv.org/abs/q-bio/0402015"
},
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