The complex geometry of the Kowalewski-Painlevé analysis
type: Item
instance of: scholarly article
title: The complex geometry of the Kowalewski-Painlevé analysis
zbMATH Open document ID: 0678.58020
DOI: 10.1007/BF01850654
author: Mark Adler
author: Pierre van Moerbeke
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143692
review text: An integrable system is algebraic complete integrable if its trajectories are straight line motions on complex algebraic tori. Improving a hundred years old result due to S. Kovalevskaya the authors establish conditions for algebraic complete integrability. The main theorem states that if the system \(z'=f(z)\) is algebraic complete integrable with invariant tori not containing elliptic curves then it has a ``coherent tree'' of Laurent solutions, and inversely, if a regular Hamiltonian system has the sufficient number of polynomial invariants in involution with a ``coherent tree'' of Laurent solutions then it is algebraic complete integrable. Conditions are given for the convergence of formal Laurent solutions. As an example a geodesic flow on SO4 is treated in detail.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 4111637
zbMATH Keywords: integrable system
zbMATH Keywords: Laurent solutions
zbMATH Keywords: regular Hamiltonian system
zbMATH Keywords: geodesic flow
reviewed by: M. Farkas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4111637