  We present the phylogenetic quartet reconstruction method SAQ (Semi-algebraic
quartet reconstruction). SAQ is consistent with the most general Markov model
of nucleotide substitution and, in particular, it allows for rate heterogeneity
across lineages. Based on the algebraic and semi-algebraic description of
distributions that arise from the general Markov model on a quartet, the method
outputs normalized weights for the three trivalent quartets (which can be used
as input of quartet-base methods). We show that SAQ is a highly competitive
method that outperforms most of the well known reconstruction methods on data
simulated under the general Markov model on 4-taxon trees. Moreover, it also
achieves a high performance on data that violates the underlying assumptions.
