# Tagged Questions

Use this tag for questions related to the study of analytic functions of strictly more than one complex variables. For the single complex dimension case, use (complex-analysis).

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### Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
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### Geometric intuition for the Stein factorization theorem?

What is the intuition behind the Stein Factorization Theorem? I understand that it was originally a theorem in several complex variables, so I was wondering if there's some geometric explanation that ...
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### Uniqueness set for analytic functions of several variables

Is there a simple (and not so restrictive) condition for a set to be an uniqueness set for the space of holomorphic functions defined on some open subset $U \subseteq \mathbb{C}^n$? By uniqueness set ...
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### On the proof of Riemann extension theorem in Huybrechts

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 ...
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### Real-analytic function of two complex variables, holomorphic in first and anti-holo in second, which vanishes on the diagonal is identically zero.

The following theorem is stated as being a well-known result of the theory of several complex variables in a book I am reading (on a more or less unrelated subject): Let $f:\mathbb C^2\to\mathbb C$...
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### Questions on Quaternion Algebra (introductory stuff)

I am a relatively new Mathematics student who understands about complex numbers and how they work. I am currently trying to create a 3D computer graphics engine and I heard that quaternion algebra may ...
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### When is a quasiprojective variety Kobayashi hyperbolic?

I am looking for some (simple and well-known) sufficient conditions on a quasiprojective variety to be Kobayashi hyperbolic. I realize that in this generality it may be a complicated (maybe even ...
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### Every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy

I am trying to prove that every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy. Let $G$ be a convex domain. I pick a point $p \in \partial{G}$. Then By Hahn-Banach Separation theorem,...
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### Riemann Mapping theorem doesn't hold true in Several Complex Variables

I need to show that Riemann Mapping Theorem is not true in general for $\mathbb{C^n}$. I know Cartan's Uniqueness Theorem and $Aut(B)$ acts transitively on $B$. But I am unable to deduce the result ...
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Suppose that $f \in H(\Omega)$ and suppose that $\Omega$ contains the closure of some polydiscs $D(p;r)$. By polydiscs I means $D(p;r)=\{z=(z_1,z_2,..,z_n) \in \mathbb{C^n}| |z_i-p_i| \lt r_i \}$ for $... 1answer 43 views ### Connectedness of$\hat{K}_U$. Let$U\subset\mathbb{C}^n$be a domain and$K\subset U$compact. If$K$is connected then$\hat{K}_U$is also connected?$\hat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \forall f\in \mathcal{O}(U)\}$:... 1answer 53 views ### Domain of holomorphy: finding a holomorphic function. Let$U\subset\mathbb{C}^n$be a domain of holomorphy. The following proposition is true? For each$a\in\partial U$, there is a holomorphic function$f\in H(U)$, such that$\displaystyle\sup_{j\ge 1} ...
Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...