An elementay proof of the automorphism theorem for the polynomial ring in two variables
type: Item
instance of: scholarly article
title: An elementay proof of the automorphism theorem for the polynomial ring in two variables
zbMATH Open document ID: 0656.13002
DOI: 10.1016/0022-4049(88)90137-5
author: James H. McKay
author: Stuart Sui-Sheng Wang
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: The authors have given an elementary proof of the following theorem: If \(\phi\) is a \({\mathbb{C}}\)-algebra automorphism of \({\mathbb{C}}[x,y]\) with \(\phi (x)=f(x,y)\), \(\phi (y)=g(x,y)\), \(\deg (f(0,y))=n\geq 1\), \(\deg (g(0,y))=m\geq 1\), then \(m| n\) or \(n| m\). From this they easily deduce that every \({\mathbb{C}}\)-algebra automorphism is a finite (compositional) product of automorphisms of the following types: \((i)\quad x\mapsto \lambda_{11}x+\lambda_{12}y+\lambda_{13},\) \(y\mapsto \lambda_{21}x+\lambda_{22}y+\lambda_{23}\) with \(\lambda_{11}\lambda_{22}-\lambda_{12}\lambda_{21}\neq 0\) and \(\lambda_{ij}\in {\mathbb{C}}\); \((ii)\quad x\mapsto x,\) \(y\mapsto y+h(x)\) with h(x)\(\in {\mathbb{C}}[x]\).
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 12D05
zbMATH DE Number: 4071046
zbMATH Keywords: automorphism of complex polynomial algebra
reviewed by: Przemysław Skibiński
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90137-5
OpenAlex ID: W2074842186
cites work: Q4097032
cites work: Homology of the group of automorphisms of \(k[x,y]\)
cites work: Q3722604
cites work: Sur les algèbres de Weyl
cites work: An Inequality for Certain Pencils of Plane Curves
cites work: Über ganze birationale Transformationen der Ebene.
cites work: On the absence of nontrivial separable forms of the affine plane
cites work: Q5622339
cites work: An inversion formula for two polynomials in two variables
cites work: Q4063564
cites work: Q5549870
cites work: Q5529113
cites work: Q5818521
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
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description: scientific article; zbMATH DE number 4071046