On algebraic relations between solutions of a generic Painleve equation
Abstract
We prove that if y" = f(y,y',t,α, β,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters α, β,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y1,..,yn,y1',..,yn') is witnessed by a pair of solutions and their derivatives (yi,yi',yj,yj'). 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.
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