Algebraic independence of generic Painlevé transcendents: PIII and PVI

Abstract

We prove that if y′′=ffalse(y,y′,t,α,β,γ,δfalse) is a generic Painlevé equation from among the classes III and VI, and if y1,…,yn are distinct solutions, then tr·degC(t)Cfalse(tfalse)false(y1,y1′,…,yn,yn′false)=2n, that is y1,y1′,…, yn,yn′ are algebraically independent over C(t). This improves the results obtained by the author and Pillay and completely proves the algebraic independence conjecture for the generic Painlevé transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C(t), provided that there are no solutions in the algebraic closure of C(t). This uses techniques and results from differential Galois theory and answers a very natural question in the theory of Riccati equations.

Bounding nonminimality and a conjecture of Borovik-Cherlin

“…In the Introduction we mentioned that if an order 1 Riccati equation δy = ay 2 + by + c over C(t), d dt has no algebraic solutions then it satisfies (C 3 ); namely, any three distinct solutions are algebraically independent over C(t). This is a theorem of Nagloo used in the study of certain Painlevé equations, see [24,Proposition 2.2]. But it can also be seen using the approach of the previous section, as we now explain.…”

“…This will suffice: that action is 3-transitive and hence any three distinct realisations will be independent. Following [24], we consider the associated second order linear homogeneous differential equation…”

“…Let V be the space of solutions to (4). It is explained in [24] how the results of [16] and the assumption that (3) has no algebraic solutions imply that the differential Galois group of (4) is SL 2 (C). One of the consequences of this is that, as SL 2 acts transitively on A 2 \ {0}, the set V \ {0} is a complete type q over F .…”

When any three solutions are independent

“…The generic type p of equation is minimal and C-internal. Nagloo [23] has shown that it does satisfy D 3 . (Another argument is by [5, §4.3] where we show that the binding group acts 3-transitively on p.) 3.1.…”

Generic differential equations are strongly minimal

“…Our approach to issue (1) follows a familiar general strategy of reducing certain problems for nonlinear differential equations to related problems for associated linear differential equations. For instance, [28] applies a strategy of this nature to establish results around the Zilber trichotomy, while [4,16] use this strategy to establish irreducibility of solutions to automorphic and Painlevé equations using certain associated Riccati equations. Our technique fits into this general framework and relies on Kolchin's differential tangent space, which will provide the linear equations associated with the original nonlinear differential variety V .…”