Regular coordinates and reduction of deformation equations for Fuchsian systems
Volume 97 / 2012
Abstract
For a Fuchsian system $$ \frac{dY}{dx}=\Bigl(\sum_{j=1}^p\frac{A_j}{x-t_j}\Bigr)Y, \tag*{(F)} $$ $t_1,t_2,\dots,t_p$ being distinct points in $\mathbb{C}$ and $A_1,A_2,\dots,A_p\in{\rm M}(n\times n;\mathbb{C})$, the number $\alpha$ of accessory parameters is determined by the spectral types $s(A_0),s(A_1),\dots,s(A_p)$, where $A_0=-\sum_{j=1}^pA_j$. We call the set $z=(z_1,z_2,\dots,z_{\alpha})$ of $\alpha$ parameters a regular coordinate if all entries of the $A_j$ are rational functions in $z$. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues of the matrices $A_j$. Since a regular coordinate is a set of unknowns of the deformation equation for (F), this process gives a reduction of deformation equations. As an example, a reduction of the Garnier system to Painlevé VI is described in this framework.
