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Abstract. We prove that if y ′′ = f (y, y ′ , t, α, β, ..) is a "generic" Painlevé equation (that is, an equation in one of the families P I − P V I but with the relevant complex parameters α, β, .. algebraically independent), and if y 1 , ..., y n are solutions such that y 1 , y ′ 1 , y 2 , y ′ 2 , ..., y n , y ′ n are algebraically dependent over C(t), then already for some 1 ≤ i < j ≤ n, y i , y ′ i , y j , y ′ j are algebraically dependent over C(t). The proof combines results by the Japanese school on "irreducibility" of the Painlevé equations, with the trichotomy theorem for strongly minimal sets in differentially closed fields.

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.

Abstract. We show that for α ∈ 1/2 + Z, the second Painlevé equation P II (α) : y ′′ = 2y 3 +ty +α is geometrically trivial, that is we show that if y 1 , ..., y n are distinct solutions such that y 1 , y ′ 1 , y 2 , y ′ 2 , . . . , y n , y ′ n are algebraically dependent over C(t), then already for some 1 ≤ i < j ≤ n, y i , y ′ i , y j , y ′ j are algebraically dependent over C(t). This extend to the non generic parameters the results in [8] for P II (α).

In this paper we show that generic Painlevé equations from different families are orthogonal. In particular, this means that there are no general Backlund transformations between Painlevé equations from the different families P I − P VI .