On topological types of polynomial mappings
Denote by P(n,p,k:\({\mathbb{K}})\) for \({\mathbb{K}}={\mathbb{R}}\) or \({\mathbb{C}}\) the set of polynomial mappings \(f:{\mathbb{K}}^ n\to {\mathbb{K}}^ p\) of degree at most k with coefficients in \({\mathbb{K}}\) and by \(P_ 0(n,p,k:{\mathbb{K}})\) the set of polynomial map germs: \(f:({\mathbb{K}}^ n,0)\to({\mathbb{K}}^ p,0)\) of degree at most k with coefficients in \({\mathbb{K}}\). One says that f,\(g\in P(n,p,k:{\mathbb{K}})\) (resp. \(f,g\in P_ 0(n,p,k:{\mathbb{K}}))\) are topologically equivalent if there are homeomorphisms \(\phi:{\mathbb{K}}^ n\to {\mathbb{K}}^ n\) and \(\psi:{\mathbb{K}}^ p\to {\mathbb{K}}^ p\) (resp. germs of homeomorphisms \(\phi:({\mathbb{K}}^ n,0)\to({\mathbb{K}}^ n,0)\) and \(\psi:({\mathbb{K}}^ p,0)\to({\mathbb{K}}^ p,0))\) such that \(f{\mathbb{O}}\phi =\psi {\mathbb{O}}g\). Denote the set of topological equivalence classes of elements of P(n,p,k:\({\mathbb{K}})\) (resp. of \(P_ 0(n,p,k:{\mathbb{K}}))\) by P(n,p,k:\({\mathbb{K}})/top\) (resp. \(P_ 0(n,p,k:{\mathbb{K}})/top)\). Thom showed that P(3,3,12:\({\mathbb{R}})/top\) is an infinite set, Fukuda proved that P(n,1,k:\({\mathbb{K}})/top\) and \(P_ 0(n,1,k:{\mathbb{K}})/top\) are finite for \({\mathbb{K}}={\mathbb{R}}\) or \({\mathbb{C}}\). Sabbah showed that P(2,p,k:\({\mathbb{C}})/top\), \(P_ 0(2,p,k,{\mathbb{C}})/top\) are finite (for \(P_ 0\) the result for \(p=2\) had been obtained before by Aoki). In this paper the author shows that P(n,p,k:\({\mathbb{K}})/top\), \(P_ 0(n,p,k:{\mathbb{K}})/top\) are infinite sets if n,p,\(k\geq 3\) or \(n\geq 3\), \(p\geq 2\), \(k\geq 4\). To obtain this result the author associates to polynomial map germs f some compact sets valued map germs M(f) such that M(f) and M(g) are conjugate if f,g are equivalent (conjugation is defined by the commutativity of a diagram associated naturally to M(f) and M(g)). The proof of the theorem is then reduced to the proof of the existence of infinitely many conjugacy classes.
scientific article; zbMATH DE number 3842798