From Huybrechts:
$\textbf{Proposition: Weierstrass Prep Theorem}$
Let $$f: B_\epsilon(0) \rightarrow \mathbb{C}$$ be holomorphic on the polydisk $B_\epsilon(0)$. Assume $f(0)=0$ and $f_0(z_1) \neq 0$. Then there exists a (unique) Weierstrass Polynomial $$g(z_1,w)=g_w(z_1)$$ and a holomorphic function $h$ on some smaller polydisk $B_{\epsilon'}(0) \subset B_\epsilon(0)$ such that $$f = g \cdot h$$ (Also, here he means function product right cause composition makes no sense) and $h(0) \neq 0$.
$\textbf{Proof}$ Let $f$ be given with multiplicity $d$ and zeroes $$a_1(w),...,a_d(w).$$ Since $f_0$ is not identically zero, we can find an $\epsilon_1 >0$ such that $f_0 \in \overline{B_{\epsilon_1}(0)}$ vanishes only in $0$. Then choose $\epsilon_2,...,\epsilon_n >0$ such that $$f(z_1,z_2,...,z_n) \neq 0$$ for $\vert z_1 \vert = \epsilon_1$ and $\vert z_j \vert < \epsilon_j$ for $j=2,...,n$. Note if $w=0$ then $$a_1(0)=...=a_d(0)=0.$$ Next, it would be nice if we knew if there was a relationship between $d$ and $w$. That is, if $d$ depends on $w$ or not. The following polynomial has the same zeroes (with multiplicities) as $f_w$. $$g_w(z_1):= \prod_{j=1}^d (z_1-a_j(w)).$$ Thus, for a fixed $w$, the function $$h_w(z_1):=\frac{f_w(z_1)}{g_w(z_1)}$$ is holomorphic in $z_1$. We must now only show $g_w(z_1),h_w(z_1)$ are holomorphic in $w$. To see this, note that the coefficients of $g_w(z_1)$ can be written as $$\sum_{j=1}^d a_j(w)^k$$ for $k=1,2,...,n$.
So far I am ok, I get confused where he introduces notation $w(z_1)$ and says that $g- w(z_1)$ is holomorphic in $w$ This is the next proof step.
Thus $g-w(z_1)$ is holomorphic in $w$ if the sum of the coefficients of $g$ are holomprihc in $w$.
What does he mean by $w(z_1)$ and also, by "holomorphic IN $a$" does this mean with respect to $a$ and in treating $a$ as our variable here. I get confused when he says "holomorphic in". Sorry just need some of the notation cleared up. I am putting together some informal notes on several complex variables, in the graduate section of my webpage. Sorry in advance if this is a silly question, but I only have a Masters in math and am applying for PhDs this year in several complex so I am getting a head start. Also, if you see any errors, do feel free to edit! I am a student learner, I am new to the world of several complex. Unpopular opinion: Several complex variables is cool. ALSO WHY IS $h$ holomorphic in $z_1$? not meromorphic?