Image Milnor number formulas for weighted-homogeneous map-germs

TL;DR

This paper derives explicit formulas for the image Milnor number of weighted-homogeneous map-germs in dimensions four and five, extending previous results and proposing a method applicable to higher dimensions under certain conjectures.

Contribution

It provides new formulas for the image Milnor number in higher dimensions and introduces an interpolative approach based on existing results.

Findings

Formulas for n=4 and 5 dimensions derived

Method recovers known formulas for n=2 and 3

Potential extension to n≥6 under conjecture

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 $n\geq 6$, the method is valid as long as certain multi-singularity conjecture holds.