The geometry of the asymptotics of polynomial maps
type: Item
instance of: scholarly article
title: The geometry of the asymptotics of polynomial maps
zbMATH Open document ID: 0922.13016
DOI: 10.1007/BF02780321
author: Ronen Peretz
published in: Israel Journal of Mathematics
publication date: 1999-01-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 1195640
zbMATH Keywords: exotic surface
zbMATH Keywords: asymptotic value of polynomial map
zbMATH Keywords: jacobian conjecture
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1195640