Regular coordinates and reduction of deformation equations for Fuchsian systems

Yoshishige Haraoka

Banach Center Publications, Tome 97 (2012), p. 39-58 / Harvested from The Polish Digital Mathematics Library

Résumé

For a Fuchsian system $d Y / d x = (\sum_{j =}^{p} (A_{j}) / (x - t_{j})) Y$ , (F) $t ₁, t ₂, . . ., t_{p}$ being distinct points in ℂ and $A ₁, A ₂, . . ., A_{p} \in M (n \times n; ℂ)$ , the number α of accessory parameters is determined by the spectral types $s (A ₀), s (A ₁), . . ., s (A_{p})$ , where $A ₀ = - \sum_{j = 1}^{p} A_{j}$ . We call the set $z = (z ₁, z ₂, . . ., z_{α})$ 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.

Publié le : 2012-01-01

Zbl 1267.34155

EUDML-ID : urn:eudml:doc:281989

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     author = {Yoshishige Haraoka},
     title = {Regular coordinates and reduction of deformation equations for Fuchsian systems},
     journal = {Banach Center Publications},
     volume = {97},
     year = {2012},
     pages = {39-58},
     zbl = {1267.34155},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc97-0-3}
}

Yoshishige Haraoka. Regular coordinates and reduction of deformation equations for Fuchsian systems. Banach Center Publications, Tome 97 (2012) pp. 39-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc97-0-3/