Determining the Number and Structure of Phylogenetic Invariants

The method of invariants is an important approach in biology for determining phylogenetic information which avoids the problems involving long branch lengths that plague other methods. In this paper, we verify the conjecture on the number of algebraic generators for the ideal of polynomial invariants. We also study rational and analytic invariants and prove several criteria concerning when it suffices to work with polynomial invariants.

[1]  Jean-Pierre Serre.  Géométrie algébrique et géométrie analytique , 1956 .

[2]  J. Risler.  Le théorème des zéros en géométries algébrique et analytique réelles , 1976 .

[3]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[4]  Mike A. Steel,et al.  Classifying and Counting Linear Phylogenetic Invariants for the Jukes-Cantor Model , 1995, J. Comput. Biol..

[5]  Local reality on algebraic varieties , 1974 .

[6]  J. Felsenstein,et al.  Counting phylogenetic invariants in some simple cases. , 1991, Journal of theoretical biology.

[7]  László A. Székely,et al.  A complete family of phylogenetic invariants for any number of taxa under Kimura's 3ST model , 1993 .

[8]  J. Felsenstein,et al.  Invariants of phylogenies in a simple case with discrete states , 1987 .

[9]  D Sankoff,et al.  A remarkable nonlinear invariant for evolution with heterogeneous rates. , 1996, Mathematical biosciences.

[10]  H. Munro,et al.  Mammalian protein metabolism , 1964 .

[11]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[12]  J. A. Cavender,et al.  Mechanized derivation of linear invariants. , 1989, Molecular biology and evolution.

[13]  Steven N. Evans,et al.  Constructing and Counting Phylogenetic Invariants , 1998, J. Comput. Biol..

[14]  J A Lake,et al.  A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony. , 1987, Molecular biology and evolution.

[15]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[16]  H. Cartan,et al.  Variétés analytiques réelles et variétés analytiques complexes , 1957 .

[17]  T. Jukes.  CHAPTER 24 – Evolution of Protein Molecules , 1969 .

[18]  László A. Székely,et al.  Reconstructing Trees When Sequence Sites Evolve at Variable Rates , 1994, J. Comput. Biol..

[19]  D Sankoff,et al.  Phylogenetic invariants for more general evolutionary models. , 1995, Journal of theoretical biology.

[20]  Terence P. Speed,et al.  Invariants of Some Probability Models Used in Phylogenetic Inference , 1993 .