Birkhoff invariants and effective calculations for meromorphic linear differential equations. I (Q1238231)

The authors consider systems of meromorphic linear differential equations of first order, namely (1) \(x' =A(z)x\) with \[ A(z)=z^{r-1}\sum_{0}^{\infty}A_{\nu}z^{-\nu},\; r\geq 0, A_0\neq 0 \] where the \(A_{\nu}\), \(s\) are constant matrices of dimension \(2 \times 2\) and the power series converges for \(|z| >R\). They give a complete list of canonical forms into which (1) can be reduced by a meromorphic or analytic transformation \(x=T(z)y\). They also give a procedure for calculating \(T(z)\) which reduces a given system to a canonical form. The importance of the results is that the complete information about the solution can be obtained from the computable transformation and the knowledge of solutions of some special equations. Basic results are stated as follows: (i) There exists a meromorphic transformation \(x=T(z)y\) at \(\infty\) such that \(y'=B(z)y\) with \(B(z)\) given by \[ (2) \qquad \qquad B(z)=z^{r-1}\sum_{\nu=0}^{r}B_{\nu}z^{-\nu}, \] (ii) There exists an analytic transformation \(x=T(z)y\) at \(\infty\) such that \(y'=B(z)y\) with \(B(z)\) given by (2) or \[ B(z)=z^{r-1}\sum_{\nu =0}^{r}B_{\nu}z^{-\nu}+z^{-1-k} \left[ \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right] , \] where \(k\) is a positive integer and \(B_{\nu}\) are lower triangular. Throughout the paper it is assumed that \(r=1\) and \(A_0\) has distinct eigenvalues, and a complete listing is made of canonical forms which need to be solved in order to represent the singularities of solutions of the general equations. A system of Birkhoff invariants are introduced and computed from a formal solution. Given a system of Birkhoff invariants, the authors show also how to explicitly construct a list of all special examples having this set of invariants.