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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3
votes
1answer
27 views

an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$

Is true that if an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$? Here $\Delta^n$: polydisc and $T^n$: Torus, distinguished boundary of $\Delta^n$.
0
votes
0answers
21 views

0 Non Duplicate Number Secquence 4 [closed]

ı konw the answer to we ,but l'm more cancermed with how l can derive that answerher
-2
votes
1answer
27 views

Justify with reason abot correct option [closed]

Let $f:\mathbb R^2 \to \mathbb R$ be a continuous map such that $f(x)=0$ for only finitely many values of $x$.Which of the following is true? 1.either $f(x)\le 0$ for all x or $f(x)\ge 0$ for ...
1
vote
1answer
11 views

Pluriharmonic is harmonic

I just started learning several complex variables and I'm a little bit confused. I just read that every pluriharmonic function is harmonic and I can't find any proof of that. Please help.
0
votes
0answers
7 views

Is it true that $\overline{\mbox{span}}\{z_1^n z_2^m H^2(T^2): |z_1|=|z_2|=1 \mbox{ and } m,n\in\mathbb{Z}_- \}=L^2(T^2)$?

Is it true that $\overline{\mbox{span}}\{z_1^n z_2^m H^2(T^2): |z_1|=|z_2|=1 \mbox{ and } m,n\in\mathbb{Z}_- \}=L^2(T^2)$? Here $H^2(T^2)$ is Hardy space on $T^2$.
1
vote
0answers
25 views

Compact normal family

Let $D \subset \mathbb{C}^n$ be a bounded convex domain. Let $F_j : D \times D \rightarrow D$ be a sequence of holomorphic functions such that $F_j(q,q) = q$ for all $j$. Then $\{F_j\}_j$ is a ...
1
vote
0answers
24 views

Holomorphic funtions

Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain. I have done If $fg\equiv0$ in $U$, then $f$ ...
0
votes
0answers
20 views

Complex functions with Property

Consider the functions $f, g, h:\mathbf{C}^2$ to $\mathbf{C}$ defined as follows $$f(z,w)=\frac{a}{z}+\frac{b}{w}$$ and $$g(z,w)=a+b.\frac{z}{w}$$ and $$h(z,w)=a.\frac{w}{z}+b$$ It is easily can be ...
0
votes
1answer
13 views

mixed limit of ·$ x^{-y} $ whenever x tends to $ \infty $ and $y \to 0^{+} $

what would be the 2-dimensional limit $$ x^{-y}=A $$ $ x \to \infty $ and $ y \to 0^{+} $ i have tried with my calculator and this says that the lmit approaches to 1 but i would like to have a ...
1
vote
0answers
25 views

How to apply Fubini's theorem in proof of Osgood's lemma

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots ...
1
vote
0answers
22 views

Sum of subharmonics is subharmonic (using the more general definition)

I want to prove that a sum of subharmonic is subharmonic using the following definition "Formally, the definition can be stated as follows. Let G be a subset of the Euclidean space ${\mathbb{R}}^n$ ...
3
votes
1answer
51 views

Irreducible polynomial and the zero set of its derivative

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Consider the derivative of $P,$ $D_{\mathbb{C}}P$, as a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^n.$ I have the following question: ...
0
votes
0answers
61 views

Irreducibility of a polynomial and connectedness of its zero set

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Let $Z(P)$ denotes its zero set in $\mathbb{C}^n.$ I have the following question: Does the irreducibility of $P$ imply that $Z(P)$ is ...
0
votes
1answer
10 views

A function in $H^\infty(\Delta^2)$

Can you give an example of a function $f\in H^\infty(\Delta^2)$ with $f^{-1}\in L^\infty(T^2)$ but not inner? Here $H^\infty(\Delta^2)$ is the space of all bounded analytic functions defined on ...
0
votes
0answers
24 views

An iterative sequence of complex numbers

Consider a disk at center at $(0,0)$ of radius, $r$ $B_r(0)$ in the complex plane. Let $w_1$ and $w_2$ be two complex numbers belong to the disk $B_r(0)$. Consider a scheme, ...
5
votes
1answer
237 views

The germ induced by an irreducible polynomial

Let $P\in\mathbb{C}[z_1,z_2,\ldots,z_n]$ be an irreducible polynomial. Let $a\in\mathbb{C}^n$ be such that $P(a)=0.$ Consider the germ of holomorphic functions at the point $a,$ denoted by ...
1
vote
1answer
24 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
1
vote
0answers
19 views

How to define percentage values in terms of scalar

Imagine a game in which you choose many cards with different A,B,C values. Such as : Card 1 A - 4 B - 5 C - 6 Card 2 A - 2 B - 7 C - 4 ... and so on.. To ...
0
votes
0answers
9 views

Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
1
vote
1answer
20 views

Power series in $\mathbb{C}^2$.

In complex analysis of one variable, we have the series $\sum_nz^n$ is convergent for $|z|<1$. If i consider the same in $\mathbb{C}^2$, that the series $\sum_{n_1,n_2}z_1^{n_1}z_2^{n_2}$, for ...
0
votes
0answers
20 views

Is there a simple way to solve this system of equations?

Is there a simple way to solve easily the following system of equations in the unknowns: $ x_2 , y_1 , y_2 , z_1 , z_2 \in \mathbb{C} $ depending on fixed values $​​a, b$ and $c$ in $ \mathbb{C} $ ...
1
vote
0answers
25 views

Multiplication operator is bijective?

Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective? ...
3
votes
1answer
94 views

Is the ring of entire functions coherent?

Call a commutative ring $R$ coherent if for each $n\in \{1,2,3,\cdots\}$ and each $n$-tuple $(r_1, ..., r_n)$ in $R^n$, the kernel of the map $R^n\owns (s_1, \cdots, s_n) \mapsto r_1 s_1 +\cdots + ...
1
vote
0answers
31 views

Something like the Weierstrass preparation theorem?

I'm reading Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-two Cycles, Math. Ann. 271, 31-51 (1985). Their proof of the Noether-Lefschetz theorem is based on ...
2
votes
0answers
42 views

Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
1
vote
1answer
24 views

Sanity check : zeroes of analytic function of 2 complex variables

Let $f$ be an analytic function defined on $\mathbb{C}^2$. Suppose it vanishes on a set of the form $U \times S$, where $U$ is a disk and $S$ is a countable set with an accumulation point. Is it true ...
0
votes
0answers
23 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
2
votes
2answers
87 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
1
vote
1answer
54 views

Wirtinger derivatives and conjugate

I haven't found anywhere in the literature (that's available to me, at least) a proper explanation of the following relations for a function $f \in \mathcal{C}(\Omega)$, $\Omega$ domain of ...
6
votes
0answers
49 views

Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq0$. 2) ...
0
votes
0answers
52 views

Doubts on Hartogs' lemma

I have the following version of Hartogs' Lemma (which is used to build up the proof of Hartogs' extension theorem). Let $\{\phi_\nu\}_\nu$ be a sequence of subharmonic functions which are uniformly ...
0
votes
0answers
16 views

Show that $M$ is not equivalent to $O_k$.

Assume that a model hypersurfaces is described by $$O_k=\{(z,w)\in \Bbb C^2 \mid v=|z|^k\}\tag 1$$ and a real analytic hypersurface: $$M=\left \{(z,w)\in \Bbb C^2 \mid ...
0
votes
1answer
25 views

Unclear result in characterization of subharmonic functions.

Let $\Omega$ be a region in $\mathbb{C}$ and let $\phi:\Omega \to [-\infty,+\infty)$ be an upper semicontinuous function. TFAE: \ i)$\phi$ is subharmonic in $\Omega$, ii) for any disc ...
4
votes
1answer
83 views

Polydisc is not biholomorphic to any strictly pseudoconvex domain

I want to prove the poly disc $P=\left\{z\in \mathbb{C}^2 : |z_1|<1,|z_2|<1\right\}$ is not biholomorphic to any strictly pseudo convex domain in $\mathbb{C}^2.$ Can any one provide a hint?
4
votes
0answers
64 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
2
votes
0answers
36 views

Compute $[\Lambda,\ \bar{\Lambda}]$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
1
vote
1answer
48 views

A Submanifold $M$ of $\Bbb C^N$

I have a Proposition in my book, and I write here: For every $p \in M$, with $M$ be a hypersurface in $\Bbb C^N$ the following hold. \begin{align*} \mathcal V_p &= \left \{ X \in \Bbb C ...
0
votes
0answers
26 views

Domain of boundedness of a power series

In several texts about several complex variables, like Krantz for instance, domain of convergence and domain of boundedness of a given power series are defined, and the easy result that the former is ...
0
votes
1answer
37 views

Books for a beginner (Pseudoconvex Domains)

Can anyone recommend me a book on Pseudoconvex Domains with include definitions, as well as a few examples? I have some course notes on that subject, but it's really abstract and theoretical. I want ...
1
vote
2answers
45 views

Compute the Jacobian matrix of $f(z,w)=(ze^{i\alpha},\ w)$

I have a question: Let $f$ be a holomorphic and $f(z,w)=(ze^{i \alpha},\ w)$ with $z,\ w \in \Bbb C$. Compute its Jacobian matrix. I remember that: For a holomorphic function $f$, its Jacobian ...
-1
votes
2answers
40 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
1answer
49 views

Show that we can only find $6$ biholomorphic mappings $\phi$.

I have a problem: Let $U$ be a connected open subset of a point $(0,0)$, and $$ M= \{(z,\ w) \in \Bbb C^2: \text{Im}w = |z w|^2+|z|^8+\dfrac{15}{7} |z|^2 \text{Re}z^6 \}$$ Let $$\phi:\ U \to ...
2
votes
0answers
35 views

What is the motivation of the complex analytic spaces?

I wonder about the motivations of the complex analytic spaces, nevertheless they are too complicated and difficult. Is it just a generalization of analytic subset of complex manifolds? Or, they have ...
1
vote
1answer
42 views

winding number in several complex variables

Is there any analogue of the concept of winding numbers in the theory of several complex variables? If so, can anyone provide me references for studying it?
0
votes
0answers
29 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
1
vote
1answer
32 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
0
votes
0answers
49 views

Density of rational functions in open Stein sets

Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb C^n$ whose restriction to $U$ is ...
1
vote
0answers
108 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
1
vote
1answer
65 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
1
vote
0answers
39 views

Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...