Phylogenetic ideals and varieties for the general Markov model
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The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statistics.
For an arbitrary trivalent tree, we determine the full ideal of invariants for the 2-state model, establishing a conjecture of Pachter-Sturmfels. For the $κ$-state model, we reduce the problem of determining a defining set of polynomials to that of determining a defining set for a 3-leaved tree.
Along the way, we prove several new cases of a conjecture of Garcia-Stillman-Sturmfels on certain statistical models on star trees, and reduce their conjecture to a family of subcases.
5 figures; Minor changes, including renumbering, to reflect final publication form
5 figures; Minor changes, including renumbering, to reflect final publication form