Let k be a field of characteristic zero, ,Y ) and the transcendency of the solution in tk [[t]] of the algebraic differential equation g(t, y(t)) · (∂/∂t)y(t) = f (t, y(t)). This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.

Abstract. We prove that if the order-one differential operator

We study how to minimize the number of invariants that is su‰cient for the Whitney equisingularity of a one parameter deformation of corank one finitely determined holomorphic germ f : ðC n ; 0Þ ! ðC n ; 0Þ. According to a result of Ga¤ney, these are the 0-stable invariants and all polar multiplicities which appear in the stable types of a stable deformation of the germ. First we describe all stable types, then we show how the invariants in the source and the target are related and reduce the number using these relations. We also investigate the relationship between the local Euler obstruction and the polar multiplicities of the stable types. We show an algebraic formula for the local Euler obstruction in terms of the polar multiplicities and show that the Euler obstruction is an invariant for the Whitney equisingularity.

No abstract