Abstract
Let \({Y=(f,g,h){:} \mathbb{R}^{3} \to \mathbb{R}^{3}}\) be a C 2 map and let Spec(Y) denote the set of eigenvalues of the derivative DY p , when p varies in \({\mathbb{R}^3}\) . We begin proving that if, for some ϵ > 0, \({Spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,}\) then the foliation \({\mathcal{F}(k),}\) with \({k\in \{f,g,h\},}\) made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of \({\mathbb{R}^n.}\)
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The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.
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Gutierrez, C., Maquera, C. Foliations and polynomial diffeomorphisms of \({\mathbb{R}^{3}}\) . Math. Z. 262, 613–626 (2009). https://doi.org/10.1007/s00209-008-0393-7
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DOI: https://doi.org/10.1007/s00209-008-0393-7