Topological equivalence of planar vector fields and their generalised principal part
In the space of \(C^{\infty}\) planar vector fields are considered subspaces \(V\) of vector fields with isolated not monodromic (not a center nor a focus) singularities and a fixed Newton diagram. For \(X \in V\) the author defines generalized, nondegenerate generalized and minimal generalized principal parts of the vector field by a filtration and some other properties of the Newton diagram. The main result of the paper is that \(X\) is locally topologically equivalent to the minimal generalized principal part. The method of proof is induction on \(n,\) where \(n\) is a number of useful sides (sides that are not entirely contained in \(\{x \leq 1 \} \cup \{y \leq 1 \}\)) of the Newton diagram. During induction the normal form method and blowing-up method are used. This result generalizes previous results by \textit{F. S. Berezovskaja} [Russ. Math. Surveys 33, No. 2, 227-228 (1978; Zbl 0397.34019)] and \textit{M. Brunella} and \textit{M. Miari} [J. Differ. Equations 85, No. 2, 338-366 (1990; Zbl 0704.58038)] and the result by the author [Glas. Mat., III. Ser. 31, No. 2, 333-351 (1996; Zbl 0888.32018)].
scientific article; zbMATH DE number 1529991