The geometry of the asymptotics of polynomial maps
If \(f:\mathbb K^n\to\mathbb K^n \) is a polynomial map (\(\mathbb K =\mathbb R\) or \(\mathbb C\)) then a point \(X_0\in \mathbb K^n \) is called an asymptotic value of \(f\) if there exists a curve \(\sigma (t)=(X_1(t),\dots, X_n(t))\), \(0\leq t<\infty\) such that \(\lim_{t\to\infty}| \sigma(t)| =+\infty\) and \(\lim_{t\to\infty}f(\sigma(t))=X_0\). In the paper many topics concerning the polynomial mappings and, in particular, jacobian conjecture are considered. All the topics are connected with asymptotic values. The author 1. proves that there is no counterexample to the jacobian conjecture over \(\mathbb C\) obtained by using the construction of Pinchuk's map, 2. reproves equivalent formulation of the jacobian conjecture in terms of Newton polygons (results of S. S. Abhyankar and J. Lang), 3. proves equivalent formulation of the jacobian conjecture in terms of finiteness of the mapping (well known result because the finiteness is equivalent to properness in the affine case), 4. gives examples of exotic manifolds.
scientific article; zbMATH DE number 1195640