On finite determinacy of formal vector fields

Abstract

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,