I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition).
The Weierstrass Preparation Theorem says the following.
Theorem (Weierstrass Preparation Theorem). If $f$ is holomorphic around the origin in $\mathbb{C}^n$ and is not identically zero on the $w$-axis, then in some neighbourhood of the origin $f$ can be written uniquely as $$f=g\cdot h$$ *where $g$ is a Weierstrass polynomial of degree $d$ in $w$, i.e., $g$ if of the form*$$g(z,w)=w^d+a_1(z)w^{d-1} + ... + a_d(z)$$ and $h(0)\neq 0$.
I understand the theorem and its proof, but then the authors conclude the following which I am not being able to understand.
"(Therefore) The zero locus of an analytic function $f(z_1,...,z_{n-1},w)$, not vanishing identically on the $w$-axis, projects locally onto the hyperplane ($w=0$) as a finite-sheeted cover branched over the zero locus of an analytic function."
Could someone help me elaborate a little bit further so that I can understand the meaning of this last paragraph?
I understand that as a consequence of the theorem, the zero locus of $f$ is locally (for most choices of coordinate systems) the same as the zero locus of $g$. This is evident since $h$ does not vanish around zero. So my intuition says that the sheets will be related to the zero loci of the functions $a_i$. But I am missing the geometric picture to see the branched cover.