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Página principal > Artículos > Artículos publicados > Differential Galois theory and non-integrability of planar polynomial vector fields |
Fecha: | 2018 |
Resumen: | We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function. |
Ayudas: | Ministerio de Economía y Competitividad MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-504 |
Nota: | Altres ajuts: MECC/MTM2015-65715-P |
Derechos: | Tots els drets reservats. |
Lengua: | Anglès |
Documento: | Article ; recerca ; Versió acceptada per publicar |
Publicado en: | Journal of differential equations, Vol. 264, issue 12 (June 2018) , p. 7183-7212, ISSN 1090-2732 |
Postprint 28 p, 448.6 KB |