Submission date: 13 December 2011.

Abstract:

We prove that if y” = f(y,y',t,\alpha, \beta,..) is a generic Painlevé equation (i.e. an equation in one of the families PI-PVI but with the complex parameters \alpha, \beta,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on “irreducibility” of the Painlevé equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.

Mathematics Subject Classification: 14H05, 14H70, 34M55, 03C60

Keywords and phrases:

Full text arXiv 1112.2916: pdf, ps.