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The Quarterly Journal of Mathematics Advance Access originally published online on October 19, 2006
The Quarterly Journal of Mathematics 2007 58(1):81-90; doi:10.1093/qmath/hal017
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© 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

VECTOR FIELDS IN R2 WITH MAXIMAL INDEX

A. C. Nabarro{dagger} and M. A. S. Ruas

Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil

{dagger} Corresponding author. E-mail: anaclana{at}icmc.usp.br

Received 25 October 2005; revised 21 June 2006
  Abstract

We consider the method of Poincaré to investigate the local index of vector fields in the plane. If m is the degree of the first non-zero jet, Xm, of the vector field X at an isolated zero, we explore the geometry of the pencil generated by the coordinate functions of Xm when the absolute value of the index of X, |ind (X)|, is m. We also find necessary and sufficient conditions for |ind (X)| to be m.


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