Phylogenetic invariants for more general evolutionary models

An invariantQof a treeTunder ak-state Markov model, where a generalized time parameter is identified with theEedges ofT,allows us to recognize whether data onNobserved species (usually,NDNA sequences, one from each species) can be associated with theNleaves ofTin the sense of having been generated onTrather than on any otherN-leaf tree. The form of the generalized time parameter is a positive determinant matrix in some semigroupSof Markov matrices. The invariance is with respect to the choice of the setEmatrices inS,one associated with each of theEedges ofT.The parametric form ofSrepresents a model of the evolutionary process. In this paper, we apply a general method of finding invariants of a parametrized functional form to find low-degree polynomial invariants for different models. Quadratic invariants are obtained for the Kimura two-parameter model, for a model allowing evolutionary dependence between positions in the sequences and for an asymmetric model that allows for A + T versus G + C asymmetries in DNA base composition. Those invariants are found for trees (unrooted in case of the Kimura model and rooted for the others) withN= 3 orN= 4 terminal vertices. We also find cubic invariants for a ten-parameter model withk= 4 states, for rooted trees withN= 4. In each case, we use implicit function theory to predict the number of algebraically independent invariants and then use this prediction to guide a systematic search for algebraic dependence within the set of invariants produced by our method.