On stable invariants of weakly semiquasihomogeneous functions

Let f:(Cn,0)→(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 μ(f) and the local Łojasiewicz exponent ł0(f) of f in terms of weak weights.