Generalised principal part of some planar vector fields (Q2785621)

The author investigates the topological equivalence of planar vector fields and their (generalised) principal part for a class of vector fields which includes some degenerate examples not covered by the usual equivalence theorem for non-degenerate fields of \textit{F. S. Berezovskaya} [Usp. Mat. Nauk 33, No. 2(200), 187 (1978; Zbl 0383.34008)] and \textit{M. Brunella} and \textit{M. Miari} [J. Differ. Equations 85, No. 2, 338-366 (1990; Zbl 0704.58038)]. NEWLINENEWLINENEWLINEBy firstly generalising the definition of the principal part of a planar vector field with an isolated singularity at the origin, the author is able to consider vector fields whose usual principal part is degenerate. Secondly, by considering only vector fields whose Newton diagram has only one useful side (so that only one blowing-up is needed for desingularisation), the main theorem is achieved: NEWLINENEWLINENEWLINELet \(W\) be the space of vector fields with only one useful side. There exists a codimension two set \(W \setminus W_2\) such that for all \(X \in W\setminus W_2\), \(X\) is locally topologically equivalent to its generalised principal part.NEWLINENEWLINENEWLINE(There is no need to make the usual exclusion of centres or focii because \(W\) does not contain them).