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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: An Introduction

Local irreducibility of the first Painlev\'e equation

14th September 2006

Author: Casale, G (Tokyo)


Let $K \supset \mathbb{C}(x)$ be an ordinary differential extension of fields. An order two differential equation $$y'' = E(x,y,y') \in \mathbb{C}(x,y,y')$$ is said to be reducible over $K$ if there exist two independent first integrals of the vectors field $ \frac{\partial}{\partial x} + y' \frac{\partial}{\partial y} + E(x,y,y') \frac{\partial}{\partial y'}$ in a differential extension $L$ of the partial differential field $K(y,y')$ (with derivations $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$, $\frac{\partial}{\partial y'}$) such that $K(y,y')=L_0 \subset L_1 \subset \ldots \subset L_p = L$ with intermediate extensions $L_i \subset L_{i+1}$ \begin{itemize} \item[--] algebraic extension, \item[--] strongly normal extension,

\hspace*{1cm}[ {\sl typical s.n.\,extensions are $L_{i+1} = L_i(h_1,\ldots,h_q)$ with $dh_i = \sum h_j \omega_i^j$ ; $\omega_i^j$ 1-forms with coefficients in $L_i$} ] \item[--] codimension one strongly normal extension,

\hspace*{1cm}[ {\sl there is $H \in K_i$ such that $K_{i+1} = K_i(< h_1,\ldots,h_q>)$ with $dh_i = \sum h_j \omega_i^j \mod H$; $\omega_i^j$ 1-forms with coefficients in $L_i$} ] \item[--] extension by a first integral of a codimension one foliation,

\hspace*{1cm}[ {\sl there is an integrable 1-form $\omega$ with coefficients in $L_i$ and $K_{i+1} = K_i (< h >)$ with $dh \wedge \omega = 0$} ] \end{itemize} where $(<\ >)$ stands for `differential fields generated by'.

An equation is said locally irreducible if it is irreducible over any field $K$.\\

In this talk, local irreducibility of the first Painlev\'e equation, $ y'' = 6y^2+x $, is investigated. The main tool used is the Galois groupoid of $P_1$ over $K$ defined by B. Malgrange. On gets a characterization of reducible equation : the transversal differential dimension of the Galois groupoid of a reducible equation is one. From the computation of the Galois groupoid of $P_1$, we prove that the transversal differential dimension of the Galois groupoid of $P_1$ is two. This computation involves \'E. Cartan classification on structural equation of pseudogroups on $\mathbb{C}^2$ and special weights on dependent variables following H. Umemura.