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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Representation of elements in a field

Let $A$ and $B$ be integrals domains, such that $A$ is integral over $B$. Writing $K(A)$ for the field of fractions, suppose that $K(A)$ is generated over $K(B)$ by a single element in $A$, say ...
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28 views

Analyticity of roots of a polynomial in terms of coefficients

Suppose that $f(z,w)$ is a non-constant polynomial in $z,w$ with coefficients in $\mathbb{C}$. Fix $z$, we define $p(w)=f(z,w)$. From Liouville's theorem, we know that $p(w)=0$ is solvable for $w$, ...
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2answers
45 views

Gradient Of Complex Least Squares

I want to find the gradient of $f : \mathbb{C}^{N} \rightarrow \mathbb{R}$, where $$ f(\mathbf{x}) = \frac 1 2 \Vert \mathbf{Ax} - \mathbf{b} \Vert_2^2, $$ and $\mathbf{A}\in\mathbb{C}^{M \times ...
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1answer
19 views

Analytic $F:\Omega\to\Omega'$ is proper iff $\{p_i\}\subset\Omega$ has no limit point in $\Omega$ implies $\{F(p_i)\}$ has no limit point in $\Omega'$

I'm reading Rudin's Function Theory in the Unit ball of $\mathbb{C}^n$. I've just embarked on reading Chapter 15, but right out of the gate he makes an assertion that I cannot justify: We shall ...
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1answer
43 views

Sets of uniqueness in $\mathbb{C}^2$

A set $M$ is called a set of uniqueness for functions of a class $\mathcal{F}$ if any function $f \in \mathcal{F}$, equal to $0$ on $M$, is identically equal to $0$. Prove that the following sets are ...
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1answer
38 views

Convergence Domain $\{|z|+|w| < 1\}$

Construct a power series whose domain of convergence is $\{(z, w) \in \mathbb{C}^2 : |z|+|w| < 1\}$ Having a bit of trouble with this problem, and was wondering if anyone had any ideas. There's a ...
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3answers
47 views

Necessary condition for a polynomial of several complex variables to vanish at a point

For a $1$-variable polynomial $f(x)\in\Bbb C[x]$ it is well-known that $$f(a)=0\iff f(x)=(x-a)g(x)$$ for some polynomial $g(x)\in\Bbb C[x]$. Question: Does that principle carries over to ...
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1answer
53 views

Open Mapping Theorem from $C^n$ to $C^n$

I am looking for an Open Mapping Theorem for a holomorphic function $f: U \subset \mathbb{C}^n \to \mathbb{C}^n$ where $U$ is a domain. I believe the following is true: Let $f: U \subset ...
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1answer
20 views

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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236 views

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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0answers
24 views

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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1answer
36 views

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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2answers
52 views

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 ...
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3answers
59 views

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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1answer
57 views

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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0answers
36 views

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 ...
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1answer
49 views

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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1answer
35 views

Gluing together Riemann surfaces, don't see why $Z$ is union of two compact sets.

Consider Lemma 1.7 from page 60-61 of Miranda's Algebraic Curves and Riemann Surfaces. For a link to the book, see here. Lemma 1.7. With the above construction, $Z$ is a compact surface of genus ...
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1answer
57 views

Automorphisms of $\mathbb{C}[x_1, \dots, x_n]$

Are the linear transformations, and the automorphisms of the form $\sigma(x_1, \dots, x_n) = (x_1 -f(x_2, \dots, x_n), x_2, \dots, x_n)$, where $f$ is a polynomial, generators of the group of ...
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37 views

Every Holomorphic function on a Hartog's figure can be extended holomorphically to the while of $P^n$

Let $f$ be a holomorphic function on the Euclidean Hartog's figure that is $$H=\{(z,w)\in P^2 : 1 \gt|z| \gt q_1 \text{or} |w| \lt q_2 \}$$ where $0 \lt q_i \lt 1$. I need to show that it has a ...
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21 views

global nullstellensatz

Let J be an ideal of analytic functions in several variables on the open unit ball. If Z(J), the analytic set of J (common zero set in the ball) is a compact subset of the ball does a global ...
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1answer
41 views

Proving a complex polynomial is identically $0$.

Let $f:\mathbb{C}^n\to\mathbb{C}$ be a complex polynomial of $n$ complex variables. Let $T^n:=\{(e^{i\theta_1},\dots,e^{i\theta_n}),\theta_j\in\mathbb{R}\}$ be the $n$-dimensional torus and $\sigma_n$ ...
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67 views

Zero set of an analytic function

If $f$ is a holomorphic function on $C^n$ which vanishes on $R^n$, it is easy to see that it vanishes everywhere. But if the zero set of $f$ is contained in $R^n$, can I deduce that $f$ vanishes ...
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1answer
73 views

Is every sheaf a subsheaf of a flasque sheaf?

Call a sheaf flasque if for all open sets $U \subset V$, the restriction map$$\mathcal{F}(V) \to \mathcal{F}(U)$$is surjective. Is every sheaf a subsheaf of a flasque sheaf?
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48 views

To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of ...
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37 views

To show $\bar{D}u=f$ where $u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge d\bar{\lambda}$,

Let $\Omega \subset \mathbb{C}$ be a bounded open set. Suppose $f \in C^{1 }(\Omega)$, $f$ is bounded and $$u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge ...
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90 views

Determine the Automorphism group of the unit bidisc.

I'm currently reading through Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range, and I'm stuck on Exercise E.2.4, which states Let $\Delta^2$ be ...
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vanishing theorem in algebraic geometry

This is a general question: As we know there are lot of vanishing theorem like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for ...
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49 views

entension of a holomorphic map on projective algebraic manifolds

I'm reading a paper.It says if $M$ is a projective algebraic variety and $f:N-S\rightarrow M$ is a holomorphic map,where $N$ is a connected complex manifold and $S\subset N$ is a proper analytic ...
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41 views

Boundary of the image of a holomorphic map in several complex variables

Suppose we have a bounded open set $U \subset \mathbb{C}^n$ and a surjective holomorhpic map $f: U \rightarrow V$, where $V$ is open in $\mathbb{C}^n$ and not necessarily bounded. Also suppose we have ...
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43 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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1answer
64 views

what is the Jacobian for transformation from complex coordinate to real coordinate?

I'm trying to figure out what is the Jacobian when you do this simple transformation: $dzdz^* \rightarrow dxdy$ where $z=x+iy$ and $z^*=x-iy$. Follow the formula we have $$J(x,y)= \begin{vmatrix} ...
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58 views

Understanding Cartan's Theorem (Small hiccups in the proof)

I was going through the theorem and the Proof of "Cartan's Theorem". The theorem says: $$\text{Suppose $\Omega$ is a bounded region in $\mathbb{C^n}$, $F:\Omega \to \Omega$ is holomorphic, for some ...
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40 views

Are the Elementary Binary Operations Analytic?

Are the elementary binary operations of addition, multiplication, and exponentiation -- taken as multivariate functions over the real numbers -- analytic? That is, $f(a, b) = a + b$ and so forth. Does ...
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24 views

$(P,H)$ Euclidean Hartogs figure

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$ and $f:H\to \mathbb{C}^n$ a holomorphic injective map, then we know that $f$ extend holomorphically to polidisc $P$ (i.e. there is a ...
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1answer
32 views

Complex Hessian outputs don't match

Let $\Omega:=\{(x_1,y_1,z_2,\bar z_2)\;:\;\underbrace{y_1-f(z_2,\bar z_2)+g(z_2,\bar z_2)}_{=:r(x_1,y_1,z_2,\bar z_2)}<0\}\subset\Bbb C^2$ (we confuse consciously $\Bbb C$ with $\Bbb R^2$) where ...
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12 views

$r_1 + r_2 =$ constant (satisfied by these distances) implies the relation$ T. grad(r1 + r2) = 0$,

If $r_1$ and $r_2$ denote the distances from a point $(x, y)$ on an ellipse to its foci, show that the equation $r_1 + r_2 =$ constant (satisfied by these distances) implies the relation ...
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1answer
157 views

Riemann extension theorem for dimension greater than 1 case

This is a theorem in P.Griffiths Principles of Algebraic Geometry. I do not get a condition and the relation between two functions. $C^n$ below is the complex n-dimensional space. Theorem: Suppose ...
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8 views

Coordinate change for a real hypersurface

Let $\Omega\subseteq\Bbb C^n$ be a domain, $0_{\Bbb C^n}\in\Omega$, $r\in\mathcal{C}^2(\Omega,\Bbb R)$, $r(0)=0$. Call $M$ the real hypersurface described by $r=0$ around $0\in\Omega$. By implicit ...
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36 views

GCD for real analytic functions

In the theory of real analytic functions of several variables, is there a concept of greatest common divisor. If so, does it also hold true that if the gcd of a collection of functions is $1$, then ...
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1answer
27 views

Question about analytic polyhedra

Let $\Pi\subset\subset U\subset\mathbb{C}^n$ be an analytic polyhedron $$\Pi=\{z\in U:|f_j(z)|<1,1\le j\le m\}$$ where $f_1,\ldots,f_m\in H(U)$, the following equality holds? ...
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Power Series in Poly Discs (Assuming Cauchy Integral Formula)

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 ...
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1answer
40 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 ...
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1answer
47 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} ...
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Multivariate Residue Theorem

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 $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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17 views

$U:$ domain of holomorphy, $d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$.

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, then we know that $$d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$$ for each compact subset $K\subset U$, also that ...
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2answers
61 views

Definition of a complex fiber

We define a real hypersurface as a subset $M\subset\Bbb C^n$ which is locally defined as the zero-locus of some $r\in\mathcal C^2(\Omega,\Bbb R)$ ($\Omega\subseteq\Bbb C^n$ open). Then let $z_0\in M$. ...
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85 views

The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as an ...
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28 views

change of complex variables

Suppose that $z_1 , z_2,\cdots ,z_k,\cdots ,z_d$ are complex variable numbers. Locally, and suppose $f_1, f_2 \cdots, f_k $ are $k$ holomorphic functions on $z_1, z_2 \cdots,z_d$. At ...
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holomorphic range of a zero variety

Let D be the unit disk, and $d\ge 2$. $F=(f_1,...f_d)$ is a holomorphic map over the closure of $D^d$ to $C^d$. What information can we get about the image of $\{0\}$x $D^{d-1}$ under $F$? Hope some ...