Preservation of level sets by automorphisms of \(\mathbb{C}^ n\)
For \(n\geq 1\) let \(f: \mathbb{C}^ n\to \mathbb{C}\) be a nonconstant polynomial function and \(g: \mathbb{C}^ n\to \mathbb{C}\) an arbitrary entire holomorphic function. Then \(f= g\) if and only if \(f^{-1}(0)= g^{-1}(0)\) and \(f^{-1}(1)= g^{-1}(1)\). The author gives a straightforward proof of this fact. Let \(a,b,c,d\in \mathbb{C}\), \(a\neq b\), and \(\phi: \mathbb{C}^ n\to \mathbb{C}^ n\) a biholomorphic automorphism. As an application of the above result, if \(\phi(f^{-1}(a))= g^{-1}(c)\) and \(\phi(f^{-1}(b))= g^{-1}(d)\), then \(g\circ \phi= {1\over a-b}((c- d)f+ ad- bc)\). In particular \(f\circ \phi= f\) if \(f= g\) and \(a= c\), \(b= d\). The example (for \(n=1\)) \(f(z)= e^ z\), \(\phi(z)= -z\), shows that the assumption on \(f\) to be a polynomial is essential. Finally, the author studies the case \(n=2\) in more detail. Let \(\Gamma_{rs}\), \(r\), \(s\) relatively prime positive integers, be the group of all biholomorphic automorphisms \(\phi: \mathbb{C}^ 2\to \mathbb{C}^ 2\), \(\phi= (\phi_ 1,\phi_ 2)\), with \((\phi_ 1(z,w))^ r\cdot (\phi_ 2(z,w))^ s= z^ r w^ s\). The groups \(\Gamma_{rs}\) are all isomorphic to each other except for \((r,s)= (1,1)\). Two groups \(\Gamma_{r_ 1 s_ 1}\) and \(\Gamma_{r_ 2 s_ 2}\) are conjugate in \(\Aut \mathbb{C}^ 2\) if and only if \(\{r_ 1,s_ 1\}= \{r_ 2,s_ 2\}\).
scientific article; zbMATH DE number 549288