I am reading in John Milnor's Book Singular Points of Complex Hypersurfaces and struggle to do a similar computation he did.
Namely, how can I compute explicitly the Milnor number $\mu(f)$ on the Milnor Fiber F, given by a polynomial relying on the definition with the mapping degree, i. e. the degree of the map $z \mapsto \frac{\nabla f}{\|\nabla f\|}$ on $S_{\epsilon}$, e. g. for $f(z_1,z_2,z_3,z_4):=z_1z_4+z_2^3+z_3^3+z_4^5$? (cf. https://en.wikipedia.org/wiki/Milnor_number#Geometric_interpretation)
Hereby, the degree of a mapping $f:S^n \to S^n$ is understood by means of homology, i. e. the unique number $d\in\mathbb{Z}$ with $f_*(x)=dx$, where $f_*$ is the induced function in homology.
I know that $deg(z^n)=n ~ \forall n$. Could this be helpful? Or should I first write $\frac{\nabla (f(z)}{\| \nabla f(z) \|}$ in polar form?
Also, maybe this post Number of roots the degree of the map? could be helpful. (Here, is it applicable at all despite the gradient?)