On the algebraic independence of generic Painlevé transcendents

Abstract

Abstract. We prove that if y ′′ = f (y, y ′ , t, α, β, . . .) is a generic Painlevé equation from among the classes II to V , and if y 1 , . . . , y n are distinct solutions, then tr.deg (C(t)(y 1 , y ′ 1 , . . . , y n , y ′ n )/C(t)) = 2n. (This was proved by Nishioka for the single equation P I .) For generic Painlevé VI, we have a slightly weaker result: ω-categoricity (in the sense of model theory) of the solution space, as described below.

Algebraic independence of generic Painlevé transcendents: PIII and PVI

“…For P I , the conjecture was shown to be true by Nishioka [10] using techniques from the study of differential function fields in one variable. On the other hand using model theory, Nagloo and Pillay [8] proved that the conjecture is also true in the case of the generic P II , P IV and P V . They moreover obtained a weaker result for P III and P V I : given distinct solutions y 1 , .…”

“…For , the conjecture was shown to be true by Nishioka using techniques from the study of differential function fields in one variable. On the other hand using model theory, Nagloo and Pillay proved that the conjecture is also true in the case of the generic , and . They moreover obtained a weaker result for and : given distinct solutions of generic (respectively, ) such that , then for all other solutions , except for at most (respectively, ), .…”

On transformations in the Painlevé family

“…When looking at P VI , it is more convenient to work with its hamiltonian form. 8 ([4],[8], [9], [20]). Let α = (α 0 , α 1 , α 2 , α 3 , α 4 ) ∈ C 5 be such that such that α 0 + α 1 + 2α 2 + α 3 + α 4 = 1 and let X VI (α) be the solution set in U of…”

“…[6],[8],[9],[10],[18]). Let α ∈ C and let X II (α) be the solution set in U of P II (α) y ′′ = 2y 3 + ty + α.Then 1.…”

“…[3],[8],[9],[20]). Let α, β, γ ∈ C and let X V (α, β, γ) be the solution set in U of P V (α, β, γ, −1/2)…”