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Planar vector field versions of Carathéodory's and Loewner's conjectures.

Carlos GutiérrezFederico Sánchez Bringas — 1997

Publicacions Matemàtiques

Let r = 3, 4, ... , ∞, ω. The Cr-Carathéodory's Conjecture states that every Cr convex embedding of a 2-sphere into R3 must have at least two umbilics. The Cr-Loewner's conjecture (stronger than the one of Carathéodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r ≠ ω, Cr-Carathéodory's...

On nonsingular polynomial maps of ℝ²

Nguyen Van ChauCarlos Gutierrez — 2006

Annales Polonici Mathematici

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity ( J ) : There does not exist a sequence ( p k , q k ) ² of complex singular points of F such that the imaginary parts ( ( p k ) , ( q k ) ) tend to (0,0), the real parts ( ( p k ) , ( q k ) ) tend to ∞ and F ( ( p k ) , ( q k ) ) ) a ² . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition ( J ) and if, in addition, the restriction of F to every real level set P - 1 ( c ) is proper for values of |c| large enough.

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