engleski

Generalised principal part of some planar vector fields

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 with a fixed Newton diagram. They define the principal part $X_Delta$ of a vector field $Xin Vsetminus V_1$ and prove that a vector field $Xin Vsetminus V_1$ is topologically equivalent to its principal part $X_Delta$. We consider the space $W$ of vector fields in $V$ having only one useful side in the Newton diagram and the space $W_1= Wcap V_1$ of vector fields in $W$, $W_1subset V_1$. We extend their results to some vector fields in $W_1$. We define a codimension two set $W_2subset W_1$ and the generalised principal part $X_{; ; arDelta}; ; $ of a vector field $Xin Wsetminus W_2$. We prove that $Xin Wsetminus W_2$ is topologically equivalent to its generalised principal part $X_{; ; arDelta}; ; $

vector field ; singularity ; topological equivalence ; Newton diagram ; blowing-up

nije evidentirano