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In proof of corollary 1.1.19 in Page 16. In the last paragraph he claims that

Since $g$ is irreducible and $\frac{\partial g}{\partial z_1}$ is of degree $d-1$, there exist elements $h_1,h_2\in\mathcal{O}_{\Bbb{C}^{n-1},0}[z_1]$ and $0\neq\gamma\in\mathcal{O}_{\Bbb{C}^{n-1},0}$ such that $h_1g+h_2\frac{\partial g}{z_1}=\gamma$

He says it holds from Gauss lemma, but I don't know why. Furthermore, I think there is a typo in the last sentence

If $g_w$ has a zero $\xi$ of multiplicity $>1$, then $\gamma(\xi)=h_1(\xi,w)g_w(\xi)+h_2(\xi,w)\frac{\partial g}{\partial z_1}(\xi)=0$

I think it should be $\gamma(w)$? Am I right? Thanks in advance for anyone's help.

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    Yes you are right $\gamma\in\mathcal{O}_{\Bbb{C}^{n-1},0} $ therefore it has variable $w$, for the Gauss lemma, I typically treat it as a black box coming from algebra. – yi li Dec 25 '22 at 07:45
  • If you really want to see the proof of the Gauss' lemma, see Junjiro Noguchi's wonderful book called Analytic Function Theory of Several Variables, section 2.2 page 41 – yi li Dec 25 '22 at 08:01
  • @yili Thanks a lot for you comment! –  Dec 25 '22 at 08:47

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