Tame and wild coordinates of \(\mathbb Z[x,y]\).
type: Item
instance of: scholarly article
title: Tame and wild coordinates of \(\mathbb Z[x,y]\).
zbMATH Open document ID: 1064.13007
author: Chi-Ming Lam
author: Jie-Tai Yu
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: For a commutative ring \(R\), the polynomial \(p\in R[x,y]\) is called a coordinate, if it is an image of \(x\) under some \(R\)-automorphism of \(R[x,y]\); the coordinate is tame if this automorphism is tame, i.e. a product of affine and triangular automorphisms. The known algorithms which decide whether \(p\in R[x,y]\) is a coordinate, and whether this coordinate is tame, work only in the case when \(R\) contains \(\mathbb Q\) [see the survey article of \textit{V. Drensky} and \textit{J.-T. Yu}, [in: Combinatorial and computational algebra Int. Conf. Combinatorial Comput. Algebra, 1999 Hong Kong, Contemp. Math. 264, 179--206 (2000; Zbl 0998.14026)]. In the paper under review, the authors further develop the existing machinery and give algorithms which determine whether a polynomial in two variables and with integer coefficients is a coordinate or a tame coordinate of \({\mathbb Z}[x,y]\). If the answer is positive, the algorithm provides also a corresponding automorphism. The considerations work also for polynomials over a unique factorization domain in the case of coordinates and over an Euclidean domain for tame coordinates.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 2104448
zbMATH Keywords: tame coordinates
zbMATH Keywords: wild coordinates
zbMATH Keywords: algorithm
zbMATH Keywords: automorphisms of polynomial algebras
zbMATH Keywords: tame automorphisms
zbMATH Keywords: wild automorphisms
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.08.022
OpenAlex ID: W1967080375
cites work: Q3808246
cites work: Q4700803
cites work: Automorphisms and coordinates of polynomial algebras
cites work: Tame and Wild Coordinates of 𝐾[𝑧][𝑥,𝑦]
cites work: Length 2 variables of \(A[x,y]\) and transfer
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: Q5818521
cites work: Polynomial automorphisms and Gröbner reductions
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
DOI: 10.1016/J.JALGEBRA.2003.08.022
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description: scientific article; zbMATH DE number 2104448