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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the solvability of planar complex linear vector fields


Author: François Treves
Journal: Trans. Amer. Math. Soc. 364 (2012), 4629-4662
MSC (2010): Primary 35A01, 35F05; Secondary 35D30, 35H10
Published electronically: April 12, 2012
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Abstract: The article discusses the local solvability, or lack thereof, of vector fields whose coefficients are complex-valued linear functions in $ \mathbb{R}^{2}$ (here called complex linear). It is proved that such a vector field is locally solvable in $ \mathbb{R}^{2}$ if and only if it is locally solvable and does not have compact orbits in the complement of its critical set or, equivalently, its Meziani number is different from zero.


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Additional Information

François Treves
Affiliation: 12002 Spruce Canyon Circle, Golden, Colorado 80403
Email: treves.jeanfrancois@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05429-8
PII: S 0002-9947(2012)05429-8
Received by editor(s): November 29, 2009
Received by editor(s) in revised form: July 22, 2010
Published electronically: April 12, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.