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Regular coordinates and reduction of deformation equations for Fuchsian systems

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Haraoka Y.

Banach Center Publications

2012

tom 97

nr 1

39-58

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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1 Banach Center Publications

1 Haraoka Y.

1 2012

Wyniki wyszukiwania

Regular coordinates and reduction of deformation equations for Fuchsian systems