Regular coordinates and reduction of deformation equations for Fuchsian systems

Yoshishige Haraoka. "Regular coordinates and reduction of deformation equations for Fuchsian systems." Banach Center Publications 97.1 (2012): 39-58. <http://eudml.org/doc/281989>.

@article{YoshishigeHaraoka2012,
abstract = {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.},
author = {Yoshishige Haraoka},
journal = {Banach Center Publications},
keywords = {Fuchsian system; accessory parameters; deformation equation; Painlevé VI; Garnier system},
language = {eng},
number = {1},
pages = {39-58},
title = {Regular coordinates and reduction of deformation equations for Fuchsian systems},
url = {http://eudml.org/doc/281989},
volume = {97},
year = {2012},
}

TY - JOUR
AU - Yoshishige Haraoka
TI - Regular coordinates and reduction of deformation equations for Fuchsian systems
JO - Banach Center Publications
PY - 2012
VL - 97
IS - 1
SP - 39
EP - 58
AB - 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.
LA - eng
KW - Fuchsian system; accessory parameters; deformation equation; Painlevé VI; Garnier system
UR - http://eudml.org/doc/281989
ER -