209 research outputs found
Toric ideals of homogeneous phylogenetic models
We consider the phylogenetic tree model in which every node of the tree is
observed and binary and the transitions are given by the same matrix on each
edge of the tree. We are able to compute the Grobner basis and Markov basis of
the toric ideal of invariants for trees with up to 11 nodes. These are perhaps
the first non-trivial Grobner bases calculations in 2^11 indeterminates. We
conjecture that there is a quadratic Grobner basis for binary trees. Finally,
we give a explicit description of the polytope associated to this toric ideal
for an infinite family of binary trees and conjecture that there is a universal
bound on the number of vertices of this polytope for binary trees.Comment: 6 pages, 17 figure
Toric fiber products
We introduce and study the toric fiber product of two ideals in polynomial
rings that are homogeneous with respect to the same multigrading. Under the
assumption that the set of degrees of the variables form a linearly independent
set, we can explicitly describe generating sets and Groebner bases for these
ideals. This allows us to unify and generalize some results in algebraic
statistics.Comment: 19 page
Phylogenetic Algebraic Geometry
Phylogenetic algebraic geometry is concerned with certain complex projective
algebraic varieties derived from finite trees. Real positive points on these
varieties represent probabilistic models of evolution. For small trees, we
recover classical geometric objects, such as toric and determinantal varieties
and their secant varieties, but larger trees lead to new and largely unexplored
territory. This paper gives a self-contained introduction to this subject and
offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure
Constructive degree bounds for group-based models
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. We prove that
for the 3-Kimura model, corresponding to the group G=Z2xZ2, 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 , such that for any tree, the associated projective scheme can be
defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were
extended to G-models (including 2-Kimura and all JC
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