On the uniqueness of the quasihomogeneity

Abstract

The aim of this paper is to show that the quasihomogeneity of a quasihomogeneous germ with an isolated singularity uniquely extends to the base of its analytic miniversal deformation. 1. Notation. Let f : (C, 0)→ (C, 0) be a germ of an analytic function and α1, . . . , αn, d be positive integers. f is called quasihomogeneous (or weighted homogeneous) of type α = (α1, . . . , αn) of quasidegree d, if ∀t ∈ C f(tx1, . . . , txn) = tf(x1, . . . , xn). The above can be stated more geometrically. f is quasihomogeneous if it is equivariant under the C∗ action on C Ψ : C∗ × C → C, Ψ(t, x) = (tx1, . . . , txn), f(Ψ(t, x)) = tf(x). If furthermore f has an isolated singularity at the origin then its Milnor number μ is finite and there is a basis of the local algebra On/If consisting of μ monomials e1, . . . , eμ, i.e. On = If ⊕ LinC{e1, . . . eμ}. We remark that If denotes the gradient ideal of f , If = On ( ∂f ∂x1 , . . . , ∂f ∂xn ) . 1991 Mathematics Subject Classification: Primary 32S30; Secondary 14B07. The paper is in final form and no version of it will be published elsewhere.