Image Milnor number formulas for weighted-homogeneous map-germs

Abstract

We give formulas for the image Milnor number of a weighted-homogeneous map-germ $({\mathbb{C}}^n,0)\to ({\mathbb{C}}^{n+1},0)$, for $n=4$ and $5$, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for $n=2$ and $3$ due to Mond and Ohmoto, respectively. For $n\ge 6$, the method is valid as long as certain multi-singularity conjecture holds.