The Jacobian conjecture in two variables
type: Item
instance of: scholarly article
title: The Jacobian conjecture in two variables
zbMATH Open document ID: 0571.13005
DOI: 10.1016/0022-4049(85)90099-4
author: Harry Appelgate
author: Hironori Onishi
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: The authors prove a special case of the Jacobian conjecture: if \(F=(f,g): {\mathbb{C}}^ 2\to {\mathbb{C}}^ 2\) is a polynomial map whose Jacobian determinant is a nonzero constant, then F is invertible, provided the degree of f or g is a product of at most two primes. The proof uses and extends ideas of \textit{Y. Nakai} and \textit{K. Baba} concerning weighted graduations on the polynomial ring \({\mathbb{C}}[X,Y]\) [cf. Osaka J. Math. 14, 403-409 (1977; Zbl 0373.12010)]. \{Reviewer's remark: Part of the Lemmas used in the proof can also be found (with different proofs) in \textit{S. S. Abhyankar}'s Tata Lect. Notes 57 (''On expansion techniques in algebraic geometry'', Bombay 1977).\}
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 3910520
zbMATH Keywords: Jacobian conjecture
Wikidata QID: Q123362561
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90099-4
OpenAlex ID: W2076018358
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: On polynomial Solutions of a differential equation
cites work: A generalization of Magnus' theorem
cites work: The amalgamated free product structure of \(\mathrm{GL}_2(k[X_1,\dots ,X_n ])\) and the weak Jacobian theorem for two variables
description: scientific article; zbMATH DE number 3910520