On commuting polynomial automorphisms of \(\mathbb C^2\)
According to a well-known theorem of \textit{H. W. E. Jung} [J. Reine Angew. Math. 184, 161--174 (1942; Zbl 0027.08503)] and \textit{W. van der Kulk} [Nieuw. Arch. Wiskd., III. Ser. 1, 33--41 (1953; Zbl 0050.26002)], the group \(\mathrm{Aut}(\mathbb{C}^2)\) of the polynomial automorphisms of the plane \(\mathbb{C}^2\) is the amalgamated product of the affine group \(A\) and the elementary group (or de Jonquières group) \(E\), over their intersection. An element of \(\mathrm{Aut}(\mathbb{C}^2)\) is said to be regular if its unique base point at the infinity and the one of its inverse are different. Using the Jung-Van der Kulk's theorem, \textit{S. Lamy} [Enseign. Math., II. Sér. 48, No. 3--4, 291--315 (2002; Zbl 1044.14035)] proved the following result: 1. If \(f\in \mathrm{Aut}(\mathbb{C}^2)\) is a regular automorphism, then the group of elements of \(A\) that commute with \(f\) is a finite cyclic group. 2. If \(f,g \in \mathrm{Aut}(\mathbb{C}^2)\) are two regular commuting automorphisms, there exists two positive integers \(n,m\) such that \(f^n\circ g^{-m}\) is an affine automorphism. The author gives new proofs of the two results, without using the Jung-Van der Kulk's theorem, whose analogue is not available in higher dimension, and hopes that his approach will give the centralizer of a regular automorphism in higher dimension.
scientific article; zbMATH DE number 2074337