Painleve Equations, Coverings of the Sphere and Belyi Functions
Davide Masoero (GFMUL)
The six Painleve equations are "integrable" nonlinear ODE in the complex plane. They are integrable in the sense that solutions only develop pole-singularities.
The main interest lies in the computation and classification of poles of solutions.
In the case of Painleve first equation, poles are classified by infinitely-sheeted branched coverings of the sphere. The pole is then computed through the holomorphic uniformization of the coverings, that is a meromorphic function with a prescribed monodromy representation.
We tackle the problem of constructing these coverings as the limit of a sequence of finitely-sheeted coverings, i.e. we approximate the sought meromorphic function by rational functions.
For a special solution of Painleve first equation, these rational functions are ramified over three points: they are Belyi functions and can be defined over some number fields.
We show how they can be explicitly constructed.