On finite determinacy of formal vector fields

It is widely understood that finite determinacy plays an important role for studies of various singularities. For instance, deformation theory of singularities of map-germs depends deeply on the algebraic characterization of finitely determined map-germs. In [1], we defined finite determinacy for singularities of formal vector fields in the same way as for map-germs. Under some conditions, we gave characterizations of finitely determined formal vector fields. First, we state briefly the results of [1]. Let K be the field of real numbers R or complex numbers C. A formal vector field X is a derivation of o~= K [ x 1 , . . . , X n ' ~ i.e. X is a K-linear mapping of ~ into itself which satisfies X ( f g ) = ( X f ) g + f ( X g ) where f, ge~.~. We say that two formal vector fields X and Y are equivalent if there is a K-algebra automorphism q) of Z such that q~, Y(=~0 -1 Yq~)=X. A formal vector field X is called k-determined if any formal vector field Y with the same k-jet as X is equivalent to X. A formal vector field X is called finitely determined if there is a positive integer k such that X is k-determined. A k-jet z is called wild if for any formal vector field X such that the k-jet of X equals z, X is not finitely determined. Now, we state several conditions on the eigenvalues. For a 1-jet X,