Generalised principal part of some planar vector fields

Vesna Županović, Zagreb, Croatia

Keywords: vector field, singularity, topological equivalence, Newton diagram, blowing-up AMS Subject classification: 58F30

Abstract: In this paper the problem of the principal part of planar vector fields is considered. In previous papers, Berezovskaya [B] and Brunella and Miari [BM] define an algebraic codimension one set V_{1} in the space V of all polynomial planar vector fields. They define the principal part X_{Δ} of a vector field X ∈ V \V_{1} and prove that a vector field X ∈ V \ V_{1} is topologically equivalent to its principal part X_{Δ}. We consider the space W of vector fields in V having only one useful side in the Newton diagram and the space W_{1} = W ∩ V_{1} of vector fields in W, W_{1} ⊂ V_{1}. We extend their results to some vector fields in W_{1}. We define a codimension two set W_{2} ⊂ W_{1} and the generalised principal part XΔ̄ of a vector field X ∈ W \ W_{2}. We prove that X ∈ W \ W_{2} is topologically equivalent to its generalised principal part XΔ̄.