Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / Online First articles

Abstract

Let $f : (\mathbb C^n, 0) \rightarrow (\mathbb C, 0)$ be a weakly semiquasihomogeneous function (i.e. the weights of $f$ are allowed to be arbitrary real numbers). We show that $f$ is right equivalent to a genuine semiquasihomogeneous function. Moreover, there is a close connection between the weights of the two. Consequently, stable-equivalence invariants expressible in terms of weights of a semiquasihomogeneous function can also be calculated in a similar way in the “weak case”. We illustrate this possibility by giving formulas for the Milnor number $\mu (f)$ and the local Łojasiewicz exponent ł$_0(f)$ of $f$ in terms of weak weights.