Partial properness and the Jacobian conjecture
type: Item
instance of: scholarly article
title: Partial properness and the Jacobian conjecture
zbMATH Open document ID: 0858.14006
DOI: 10.1016/0893-9659(96)00002-X
author: Yanyan Li
published in: Applied Mathematics Letters
publication date: 1996-10-16T00:00:00Z
review text: The author, using his own results [ibid. 9, No. 1, 27-32 (1996; Zbl 0857.57022)], proves the following theorem on injectivity of regular maps: Let \(L,M\) and \(N\) be connected complex algebraic manifolds, \(M,N\) simply connected, and \(h=(f,g)\): \(N \mapsto L \times M\) be a regular local homeomorphism. If \(\lambda\) is a connected component of a fiber \(f^{-1}(u)\), \(u \in L\), then \(h\) is a homeomorphism if and only if \(g|\lambda\) is proper and \(f\) is proper on all the connected components of fibers of \(g\) that intersect \(\lambda\). The author also gives applications to the two-dimensional Jacobian conjecture \((L=M=N=\mathbb{C})\) for \(F =(P,Q)\): \(\mathbb{C}^2 \mapsto \mathbb{C}^2\), \(P,Q\) polynomials, \(\text{Jac} (F) \equiv 1\). One of them is a generalization of the result by \textit{J. Chadzyński} and the reviewer [Bull. Soc. Sci. Lett. Łódź, Sér. Rech. Déform. 14(1992/1993), No. 131-140, 13-19 (1993)] that if \(Q\) is proper on a fiber \(P^{-1} (u)\), \(u\in \mathbb{C}\), then \(F\) is a polynomial automorphism. The author shows that it suffices to assume that \(Q\) is proper on only one connected component of \(P^{-1} (u)\) (the last statement can be reduced to the above mentioned result because one can easily show that \(Q\) must be proper on the remaining connected components).
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 915066
zbMATH Keywords: injectivity of regular maps
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: polynomial automorphism
Wikidata QID: Q123134985
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: A geometric approach to the Jacobian Conjecture in ℂ²
cites work: Q4347786
cites work: Global univalence of partially proper local homeomorphisms
cites work: Q5554916
cites work: Partial properness and real planar maps
cites work: A counterexample to the strong real Jacobian conjecture
cites work: Q3784238
cites work: On linear differential operators related to the Jacobian conjecture
cites work: On the density of image of differential operators generated by polynomials
cites work: On the Jacobian Conjecture
cites work: Q3810106
cites work: Injectivity on one line
cites work: Q4128900
cites work: Q4143433
cites work: A condition for a polynomial map to be invertible
cites work: Injective endomorphisms of varieties and schemes
cites work: Q4097032
full work available at URL: 0893-9659(96)00002-x
OpenAlex ID: W2099377585
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description: scientific article; zbMATH DE number 915066