Summary: PHYLOGENETIC IDEALS AND VARIETIES FOR THE
GENERAL MARKOV MODEL
ELIZABETH S. ALLMAN AND JOHN A. RHODES
Abstract. The general Markov model of the evolution of bio-
logical sequences along a tree leads to a parameterization of an
algebraic variety. Understanding this variety and the polynomi-
als, 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 in-
variants for the 2-state model, establishing a conjecture of Pachter-
Sturmfels. For the -state model, we reduce the problem of de-
termining 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.
An important problem arising in modern biology is that of sequence-
based phylogenetic inference. Suppose we obtain a collection of biolog-
ical sequences, such as genomic DNA, from currently extant species, or