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arXiv · 1112.2916

On algebraic relations between solutions of a generic Painleve equation

Ronnie Nagloo·Anand Pillay
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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 (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.

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Ronnie Nagloo, Anand Pillay. 2011-12-13. On algebraic relations between solutions of a generic Painleve equation. https://doi.org/10.1515/crelle-2014-0082

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