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Morphisms and inverse problems for Darboux integrating factors

Published online by Cambridge University Press:  03 December 2013

Jaume Llibre ,
Chara Pantazi  and
Sebastian Walcher
Jaume Llibre
Affiliation:
Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain ([email protected])
Chara Pantazi
Affiliation:
Department de Matemàtica Aplicada I, Universitat Politècnica de Catalunya (EPSEB), Avinguda Doctor Marañón 44–50, 08028 Barcelona, Catalonia, Spain ([email protected])
Sebastian Walcher
Affiliation:
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany ([email protected])
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Abstract

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 6 , December 2013 , pp. 1291 - 1302
Copyright
Copyright © Royal Society of Edinburgh 2013 

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