Distance to the stochastic part of phylogenetic varieties

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by hidden Markov process. This allows to define distribution characters at leaves trees and one might be able obtain polynomial relationships among probabilities different characters. The study these polynomials geometry algebraic varieties defined them can used reconstruct trees. However, not all points in have biological sense. In this paper, we explore extent which adding semi-algebraic conditions arising from restriction parameters with statistical meaning improve existing methods reconstruction. To end, our aim compute distance data stochastic part varieties. Computing distances involves optimization nonlinear programming algorithms. We use analytical find some for quartet evolving under Kimura 3-parameter or Jukes-Cantor models. Numerical computational algebra play also fundamental role paper.