Jacobian Conjecture and semi-algebraic maps
The famous Jacobian conjecture says that every polynomial local diffeomorphism \(f: \mathbb C^n \to \mathbb C^n\) is bijective (see popular surveys of \textit{A. van den Essen} [Polynomial automorphisms and the Jacobian conjecture. Basel: Birkhäuser (2000; Zbl 0962.14037)] and \textit{H. Bass} et al. [Bull. Am. Math. Soc., New Ser. 7, 287--330 (1982; Zbl 0539.13012)]). For such a map \(f\) there is a complex algebraic hypersurface \(D \subset \mathbb C^n\) away from which \(f\) is a covering map of finite degree. The Jacobian conjecture is known to hold if \(D\) is well-behaved (see work of the [\textit{S. Nollet} and \textit{F. Xavier}, Arch. Math. 89, No. 5, 385--389 (2007; Zbl 1135.14049) and J. Reine Angew. Math. 627, 83--95 (2009; Zbl 1161.14042)]). Since covering maps are proper, this motivates the stronger real Jacobian conjecture of Jelonek, which says that if \(F: \mathbb R^n \to \mathbb R^n\) is a polynomial local diffeomorphism and \(\text{codim}(S_F) \geq 2\), then \(F\) is bijective, where \(S_F \subset \mathbb R^n\) being the smallest subset for which the restriction \(F:\mathbb R^n - F^{-1}(S_F) \to \mathbb R^n - S_F\) is proper [\textit{Z. Jelonek}, Math. Z. 239, No. 2, 321--333 (2002; Zbl 0997.14017)].NEWLINENEWLINEIn the paper under review, the authors prove Jelonek's conjecture for semi-algebraic local diffeomorphisms \(F: \mathbb R^n \to \mathbb R^n\) satisfying an addition hypothesis. To state it, let \({\mathcal F}_i\) be the foliation whose leaves are the pre-images of points under the composition \(F \circ p_i\) with \(p_i\) the \(i\)th projection map. For any \(\{i_1, i_2, \dots, i_{n-2}\} \subset \{1,2, \dots, n\}\) the intersection of leaves gives rise to the foliation \({\mathcal F}_{i_1} \cap {\mathcal F}_{i_2} \cap \dots \cap {\mathcal F}_{i_{n-2}}\) of codimension \(n-2\). If the leaves of these foliations are simply connected for all such subsets, then the conjecture holds. This extends results of \textit{C. Gutierrez} and \textit{C. Maquera} [Math. Z. 262, No. 3, 613--626 (2009; Zbl 1171.37014)] and of \textit{C. Maquera} and \textit{J. Venato-Santos} [Bull. Braz. Math. Soc. (N.S.) 44, No. 2, 273--284 (2013; Zbl 1277.14047)]. For comparison, \textit{E. C. Balreira} showed that \(F\) is bijective if the pre-image of every hyperplane parallel to a fixed line is acyclic or empty [Comment. Math. Helv. 85, No. 1, 73--93 (2010; Zbl 1194.58034)].
scientific article; zbMATH DE number 6365829