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Is there an analogue of the Bezout's Theorem which pertains to holomorphic functions of two variables defined on a fixed domain or germs of analytic functions of more than one variable at $0$?

I have never seen a proof nor have I seen a counter-example.

Edit: More concretely, if we consider the ring of germs of analytic functions in several variables at $0$, is it true that for two such germs having no common factors, the set of their common zeros is discrete?

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  • $\begingroup$ Isnt this just bezout's theorem focused on a small open subset of $\mathbb{P}^1_{\mathbb{C}}$, so we just have the inequality where the sum of multiplicities is $\leq$ the product of degrees? $\endgroup$
    – basket
    Oct 20, 2016 at 15:44
  • $\begingroup$ I mean $\mathbb{P}^n _{\mathbb{C}}, n > 1$. $\endgroup$
    – basket
    Oct 20, 2016 at 16:06
  • $\begingroup$ @basket I am sorry, but I don't know about Bezout's Theorem for $\mathbb{P}^n_{\mathbb{C}}$. I only know about the Bezout's Theorem for polynomials. Does it say something about holomorphic functions of several variables also? $\endgroup$ Oct 20, 2016 at 16:16
  • $\begingroup$ I'm sorry you're right, somehow I conflated holomorphic and rational $\endgroup$
    – basket
    Oct 20, 2016 at 16:41

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