Singularities of vector fields on

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Abstract
In his well known paper `Singularities of vector fields', Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions he had to use the notion of `weak--equivalence'. In this paper we show how to classify singularities of vector fields on up to codimension 4 for the notion of equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak--equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are , although it will be clear that the results are valid for , with r sufficiently big. We provide a complete, but short, survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.