Geodesic
Schlesinger transformations for algebraic Painlevé VI solutions
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 26, Tome 487 (2019), pp. 106-139 Cet article a éte moissonné depuis la source Math-Net.Ru

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Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric $2\times 2$ system of ODEs to obtain $RS^2_4$-pullback transformations to isomonodromic $2\times 2$ Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This article demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations. 
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R. Vidunas; A. V. Kitaev. Schlesinger transformations for algebraic Painlevé VI solutions. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 26, Tome 487 (2019), pp. 106-139. http://geodesic.mathdoc.fr/item/ZNSL_2019_487_a7/

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