Reduced Forms of a Singular Vector Field in C 3 with Nilpotent Linear Part

Abstract

We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X))partial derivative(x) + (z + b(X))partial derivative(y) + c(X)partial derivative(z), where X = (x, y, z) is an element of C-3. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.