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Zeroes of complete polynomial vector fields

Author: Alvaro Bustinduy
Journal: Proc. Amer. Math. Soc. 131 (2003), 3767-3775
MSC (2000): Primary 34M45; Secondary 32S65, 14H37
Published electronically: February 26, 2003
MathSciNet review: 1998184
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Abstract: We prove that a complete polynomial vector field on $\mathbb{C} ^{2}$ has at most one zero, and analyze the possible cases of those with exactly one which is not of Poincaré-Dulac type. We also obtain the possible nonzero first jet singularities of the foliation $\mathcal{F}_X$ at infinity and the nongenericity of completeness. Connections with the Jacobian Conjecture are established.

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Additional Information

Alvaro Bustinduy
Affiliation: Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, Ciudad Universitaria 28040 Madrid, Spain

Keywords: Complete vector field, complex orbit, holomorphic foliation
Received by editor(s): January 30, 2002
Received by editor(s) in revised form: April 15, 2002, and July 1, 2002
Published electronically: February 26, 2003
Additional Notes: This paper was partially supported by a grant from Universidad Complutense de Madrid, European project TMR “Singularidades de ecuaciones diferenciales y foliaciones" and CONACYT 28492-E
Dedicated: Dedicated to my father
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2003 American Mathematical Society