Tame and wild coordinates of \(\mathbb Z[x,y]\).
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.
scientific article; zbMATH DE number 2104448