Nilpotent Singularities in Generic 4-Parameter Families of 3-Dimensional Vector Fields

This paper deals with singularities of vector fields in 3having a 1-jet linear conjugate toy(∂/∂x)+z(∂/∂y). They first occur in generic 3-parameter families. In codimension?3 all such singularities are mutuallyC0equivalent. We give a proof of this, provide a good normal form for 3-parameter unfoldings, and show that all non-wandering behaviour in such an unfolding is of small amplitude. We also show that for codimension?4 there are exactly 5 types of singularities forC0equivalence. The proof relies on normal form theory, blowing-up, and estimation of Abelian integrals.