The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
-
David Blázquez-Sanz
,
Guy Casale
Abstract
The Malgrange–Galois groupoid of Painlevé 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 the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.
Funding statement: We would like to thank the Université de Rennes 1 and the Universidad Nacional de Colombia for their hospitality and support. D. Blázquez-Sanz has been partially funded by the Colciencias project “Estructuras lineales en geometría y topología” 776-2017 code 57708 (Hermes UN 38300). G. Casale has been partially funded by the Math-AMSUD project “Complex Geometry and Foliations”. J. S. Díaz Arboleda has been partially funded by the Colciencias program 647 “Doctorados Nacionales”.
Acknowledgements
We want to thank the anonymous referee for his/her careful reading, suggestions and corrections that greatly helped to improve the quality of the paper.
References
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Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
