On algebraic relations between solutions of a generic Painleve equation

By Ronnie Nagloo, Anand Pillay. 2011

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We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve 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 Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/30f631c78b953431e321d9a7b7efc64528d9aad7