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## Banach Center Publications

2012 | 97 | 1 | 39-58
Tytuł artykułu

### Regular coordinates and reduction of deformation equations for Fuchsian systems

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For a Fuchsian system
$dY/dx = (∑_{j= }^{p} (A_j)/(x-t_j))Y$, (F)
$t₁,t₂,...,t_p$ being distinct points in ℂ and $A₁,A₂,...,A_p ∈ M(n×n;ℂ)$, the number α of accessory parameters is determined by the spectral types $s(A₀),s(A₁),...,s(A_p)$, where $A₀ = -∑_{j=1}^{p} A_j$. We call the set $z = (z₁,z₂,...,z_α)$ of α 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.
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Tom
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39-58
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2012
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• Department of Mathematics, Kumamoto University, Kumamoto, 860-8555 Japan
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