  Many of the stochastic models used in inference of phylogenetic trees from
biological sequence data have polynomial parameterization maps. The image of
such a map --- the collection of joint distributions for a model --- forms the
model space. Since the parameterization is polynomial, the Zariski closure of
the model space is an algebraic variety which is typically much larger than the
model space, but has been usefully studied with algebraic methods. Of ultimate
interest, however, is not the full variety, but only the model space. Here we
develop complete semialgebraic descriptions of the model space arising from the
k-state general Markov model on a tree, with slightly restricted parameters.
Our approach depends upon both recently-formulated analogs of Cayley's
hyperdeterminant, and the construction of certain quadratic forms from the
joint distribution whose positive (semi-)definiteness encodes information about
parameter values. We additionally investigate the use of Sturm sequences for
obtaining similar results.
