Formal implicit function theorem

Asked Oct 14 '17 at 02:59

Active Sep 25 '18 at 23:03

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I have seen a lemma about formal series in the book Complex Projective Varieties by D. Mumford.

Let $$f(x)=\sum_{i=1}^n a_iX_i+(\text{higher order terms})\in \mathbb{C}[[X]]$$ and assume $a_1\not=0$. Then $f$ can he factored as: $$f(X)=u(X_1,......,X_n)(X_1-g(X_2,......,X_n))$$ where $u(0)=0,\ g(0)\not=0$.

I have tried the case of n=2. It's quite complicated so that I can't get further.

I hope someone could give me a simple solution.

edited Sep 25 '18 at 23:03

asked Oct 14 '17 at 02:59

Intoks Liobein

Maybe Mumford, but not Zariski. – user26857 Oct 14 '17 at 18:39
I know it's a late response, but have a look at the proof of Weierstrass Preparation Theorem, you might find that quite helpful. – M.G. Sep 25 '18 at 22:36

Formal implicit function theorem

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