On the proof of Riemann extension theorem in Huybrechts

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In the proof of the generalized Riemann extension theorem in Daniel Huybrechts's Complex Geometry, which is:

Proposition 1.1.7 (Riemann extension theorem) Let $f$ be a holomorphic function on an open set $U\subset \mathbb{C}^n$. If $g:U\backslash Z(f)\to\mathbb{C}$ is holomorphic and locally bounded near $Z(f)$ , then $g$ can uniquely be extended to a holomorphic function $\widetilde{g}:U\to\mathbb{C}.$

Where $Z(f)$ is defined to be $\{ z|f(z)=0 \}$.

Before starting the rigor proof, the author had assumed that:

...We can restrict to the case that the restriction $f_0$ of $f$ to this line (which means the set $ U\cap\{ (z_1,0,\dots,0) |z_1\in \mathbb{C} \} $) vanishes only in the origin. ...

Where $f_w$ is defined by $f_w(z_1)=f(z_1,w)$.

I guess this restriction could be made under a inversible linear transformation of $\mathbb{C}^n$ to itself. But is the transformation alway exists?

Could anyone give me some details about that? Why there is no loss of generality about this?

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Yes, locally this always exists. This comes from the proof of Weierstrass Preparation Theorem.

You can always make a generic choice of a line that intersects $Z(f)$ only at $0$ up to restricting to a smaller open set $V\subset U$. After a linear change of coordinates you can assume that this line is the $z_1$-axis.