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arXiv · 2005.10291

Malgrange-Galois groupoid of Painlev\'e VI equation with parameters

David Blázquez-Sanz·Guy Casale·Juan Sebastián Díaz Arboleda
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Abstract

The Malgrange-Galois groupoid of Painlev\'e IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange-Galois groupoid of Painlev\'e VI family including all parameters as new dependent variables. We conclude it is the pseoudogroup of transformations preserving parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of Painlev\'e VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlev\'e VI.

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David Blázquez-Sanz, Guy Casale, Juan Sebastián Díaz Arboleda. 2020-05-20. Malgrange-Galois groupoid of Painlev\'e VI equation with parameters. https://arxiv.org/abs/2005.10291

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