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Questions tagged [several-complex-variables]

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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To prove that every bounded function holomorphic on $\mathbb{C}^2 \setminus K$ is constant

I have to prove that every bounded function holomorphic on $\mathbb{C}^2 \setminus K$ is constant, where $K$ is $(a)$ a ball $(b)$ a complex line $(c)$ an arbitrary analytic subset Now, I think ...
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1answer
12 views

Complex velocity of line vortex, simplification

I am considering the potential flow where I have a uniform flow past wing and two line vortices, one at the origin and one at a position $(x_1,y_1)$ relative to a wing of chord $D$. I'm using the ...
2
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1answer
28 views

Almost complex structure question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are ...
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58 views

What is the second-order Taylor expansion of a function $f : \mathbb C^n \to \mathbb R$?

Consider for example $f(x)=\|Ax-b\|_2^2$, where $A \in \mathbb C^{m \times n}$, $x \in \mathbb C^n$, $b \in \mathbb C^m$. $y := Ax - b \implies dy = A dx \implies dy^* = A^* dx^*$. Taking the ...
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1answer
37 views

Definition of holomorphic functions in multiple dimensions

What is the definition of a function $$f:U\rightarrow\mathbb{C}^n$$ being holomorphic? Where $U\subseteq\mathbb{C}^n$. When I look around online all I can see is the definition for $$f:U\rightarrow\...
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Is there an extension of the concept of Stokes lines/the Stokes phenomenon to 2 complex variables?

For one complex variable, normally we refer to the stokes(/antistokes) lines as being lines that divide the input space according to asymptotic behaviour. I hope we can extend this idea to "stokes ...
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18 views

references to boundary behaviour d-bar problem solutions

is there any reference around the subject of boundary behaviour of solutions of $\bar\partial$ problem $\bar\partial u=f$, when $f$ as slow growth at the boundary of a pseudo-convex domain $\Omega$ in ...
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2answers
62 views

Definition of divisor associated to a meromorphic function

Let $X$ be a complex manifold and let $f$ be a meromorphic function defined on it. Let $Div(X)$ be the group of locally finite sum of analytic irreducible hypersurfaxces of $X$. One would like to ...
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1answer
38 views

Does there exist a function $f:\mathbb{C}^2\to \mathbb{C}$ holomorphic aside from at singularities with isolated singularities?

All the examples I've been able to come up with are either non-holomorphic or fail the condition. For example, with the 1D case we can take $$f(z) = \frac{1}{1+z^2}$$ With singularities at $\pm i$. If ...
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2answers
23 views

Taylor series of complex multivariate function

I am looking for a reference or some literature on Taylor series of complex multivariate functions. I found material for complex functions and material for multivariate functions, but not for both. ...
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71 views

Factorization of an analytic function in $\mathbb{C}^n$

Let $\Omega$ be an open set of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one. If $...
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18 views

$\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ s.t. $Q_i$ is a prime ideal.

Prove that the ideal $\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ such that $Q_i$ is a prime ideal.
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21 views

$A\cap\Lambda$ polynomially convex in $\Lambda$

Let $A\Subset\Bbb C^n$ polynomially convex, that is its convex hull $$ \widehat A:=\{z\in\Bbb C^n\;:\;|f(z)|\le\|f\|_{A}\;\;\forall f\in\mathcal O(\Bbb C^n)\} $$ coincides with $A$. Consider then $\...
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28 views

Retrieving a function of many complex variables from its manifold of zeros

Physical background: In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The ...
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Linear functionals and complex polynomials

Question: Let $K\subset \mathbb{C}^N$ be compact and let $\mathcal{P}(K)$ denote the closure of $\{p|_K:\, p \in \mathbb{C}[z_1,\dots,z_N]\}$ with respect to $||\cdot||_K$. Show that for each non ...
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23 views

Can we extend the idea of contour independence for complex contour integrals to several complex variables?

That is, given some function $f:\mathbb{C}^n \to \mathbb{C}$ entire/ sufficiently holomorphic, if we have two domains $D,D'$ in $\mathbb{C}^n$ with the same boundary i.e $\delta D = \delta D' $, will ...
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42 views

On a sufficient condition for being a prime ideal

Consider the ideal $I$ in $\mathbb{C}[z_{1},\ldots,z_{m}]$ generated by $\{p_{1},\ldots,p_{t}\}$, where $t\leq m$ and $p_{1},\ldots,p_{t}$ intersects completely i.e. the map $(p_{1},\ldots,p_{t}):\...
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1answer
49 views

Visual representation for a function $f:\left(\mathbb{C}^2\right)^2\to\left(\mathbb{C}^2\right)^2$

I am currently working with some matrix functions $\left(\mathbb{C}^2\right)^2\to\left(\mathbb{C}^2\right)^2$, and I would like to make some sort of graphic. I had an idea to split the output into ...
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1answer
44 views

Holomorphic morphisms cannot be extended to codimension $2$

Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Given a holomorphic morphism $f:D^\times \to\mathbb P^n$, we want to know whether we can extend it to the whole ...
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Holomorphic morphism sending given curves to give points

Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Let $C_1$ and $C_2$ be two curves passing through the origin $0$, and $C_1\cap C_2=\{0\}$. We denote the ...
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0answers
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Maximum co-dimension of a submanifold given by finitely many holomorphic functions

Let $\Omega$ be a domain in $\mathbb{C}^{m}$ and $f_{1},\ldots,f_{k}:\Omega\mapsto\mathbb{C}$ are holomorphic functions. Assume that the common zero set, $Z(f_{1},\ldots,f_{k})$ is a submanifold in $\...
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27 views

Finding a holomorphic function

Question: For $z \in \mathbb{C}^{N+1}$ and $j \in \{1,\dots,N\}$, let $z(j) \in \mathbb{C}^N$ be the tuple obtained by omitting the $j^{\text{th}}$ coordinate of $z$. Let $D \subset \mathbb{C}^N$ be ...
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2answers
60 views

Links between several complex variables and number theory?

Can several complex variables be used to prove results in number theory? If there exists such a proven result, please refer me to the paper/article.. Any help would be appreciated.
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1answer
35 views

If function has a zero on an open set, then it has a zero on the boundary

Question: Let $D\subset \mathbb{C}^N$ for $N\geq 2$ be open, bounded and connected and let $f: \overline{D} \rightarrow \mathbb{C}$ be continuous such that $f$ is holomorphic on $D$. Show that if $f$ ...
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1answer
76 views

Holomorphic Morse functions

For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + ...
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0answers
50 views

Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a ...
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1answer
25 views

Example of a complex meromorphic functions which is not locally Lipschitz [closed]

Let $U\subset \mathbb{C}^n$ be an open subset and let $\varphi$ be a meromorphic function on $U$ i.e. locally given by the ratio of two holomorphic functions. Can $\varphi$ be not locally Lipschitz?
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1answer
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Does every non-compact Riemann surface embed holomorphically into $\mathbb{C}^2$?

Question: Can every non-compact Riemann surface be holomorphically embedded into $\mathbb{C}^2$? If not, what are some (all?) of the obstructions to such an embedding? This question is partially ...
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1answer
37 views

Differential of logarithm on complex plane

I will be grateful if someone could explain the following statement from p. 26 of the book "Analytic functions of several complex variables" by Gunning and Rossi: As a function of $\zeta$ for $z$ ...
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59 views

Show the zero set of a holomorphic section can be written as the union of two disjoint closed set

Let $X$ be a complex manifold and $E$ be a holomorphic vector bundle over $X$. Denote $\tilde{E}$ as the pull-back bundle of $E$ over $X\times X$ via the map $\pi: X\times X\to X$ sending $(x,y)\to x$....
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1answer
68 views

Show the set is closed

Let $X$ be a Hausdorff locally compact topological space, $s:X\times X\to \mathbb{R}$ be a continuous map such that $s(x,x)=0$ for all $x\in X$. Set $\triangle=\{(x,y): x=y, x,y\in X\}$ and $A=\{(x,y):...
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34 views

Analytic proof that infinite level sets of complex polynomials aren't compact

I've heard it stated that if you have a family $p_{1}, \ldots, p_{\ell} \in \mathbb{C}[x_1 , \ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = \bigcap_{j = 1}^{\ell} p_{j}^{-1} (\{0\...
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41 views

How to minimize $\sum_i |a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|x_2|^2 - b_i|^2$ over $x_1, x_2 \in \mathbb C$?

Consider the following nonlinear minimization problem \begin{align} \tag{P1} \min_{x_1, x_2 \in \mathbb C} \sum_{i=1}^m \big|a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|...
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1answer
29 views

roots of a several variable for the equation $y-3x^2-y^3=0$

How can I get the roots of the next equation? $$y-3x^2-y^3=0$$ I just dont get the same answer than my teacher: $$x = \frac {- \sqrt2}{3(3^{1/4})}, y = \frac{-2}{\sqrt3}$$
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1answer
69 views

Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
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1answer
79 views

Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.

Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$. Goal: Prove that $p$ is irreducible. Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by $$I:=(p).$$ My approach is to show that ...
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22 views

Simplify a symmetric complex function .

Lets consider the following symmetrical complex relation $f(z) = \frac{B(z)B(\frac{1}{z})}{B(z)B(\frac{1}{z}) + \alpha A(z)A(\frac{1}{z})}$ where $\alpha$ is a real value, $B(z)=\sum_{i=1}^{N}\...
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27 views

Factorising vector spaces

Suppose I have some complex (unit) vector $V$ on a vector space $\mathcal{H}$ of dimension $N$, where $N$ has prime factorisation $N=\prod^d_i p_i$ (multiplicity is allowed- the $p_i$ need not be ...
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1answer
34 views

Upper envelope of plurisubharmonic functions

Suppose that $\{u_\alpha\}_{\alpha \in A}$ is a family of plurisubharmonic functions (psh function) on $\Omega \subset\subset \mathbb{C}^n$. Then, let $u(z) = \sup_{\alpha \in A} u_{\alpha}(z)$ be the ...
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51 views

Maximal Natural Domain of An Analytic Function on Complex Plane

I want to ask a question of the maximal natural domain of an analytic function. We know that on $\mathbb{C}^n$, a domain $U$ is a domain of holomorphy if and only if $U$ is a pseudoconvex domain, and ...
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0answers
20 views

Finding Levi Matrix [closed]

Who represent Levi Matrix $n\times n$ in several complex variables? I read many documents and books but I can't find a Levi Matrix explicitly. Can you help me ?
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1answer
32 views

A real-analytic radial function on closed unit ball which peaks at zero is strictly decreasing

Let $\mathbb{B}$ be the closed unit ball in $\mathbb{C}^n$ and let $g:\mathbb{B}\rightarrow \mathbb{C}$ be a real-analytic radial function such that $g(0)=1$ and $|g(z)|<1\, \forall\, \text{non-...
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12 views

Some problem of complex multi-function

Assume that $f:U\rightarrow \mathbb{C}$ is a holomorphic function on a connected open subset of $\mathbb{C}^n$. Then can we prove that for every point $x\in U$. There would be a local coordinates $(...
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31 views

Extension of holomorphic function to exceptional divisors

Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it. Let $\pi:E \to \...
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0answers
20 views

On constancy of a complex multivariable polynomial

consider,a nonconstant polynomial $f(x,y,z,w)$ in $\mathbb C$ $[x,y,z,w]$ as a continuous map from $\mathbb C^4$ to $\mathbb C$. Now, if we have all four partial derivatives of $f$ vanishes at all ...
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1answer
44 views

Several variables contour integral

Suppose I have an entire function of $n$ complex variables $f \colon \mathbb C^n \to \mathbb C$ and $f$ is rapidly decreasing (or enough conditions to make the following work). I want to show that $$\...
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48 views

Nonlinear optimization with complex residual and jacobian

I am trying to minimize the following function $\chi^2=(f(t_i;\vec{p})-y_i)^{H}\text{Cov}^{-1}_{ij}(f(t_j;\vec{p})-y_j)$ where $A^H$ is the hermitian conjugate of A, $f(t;\vec{p})-y$ is a complex ...
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1answer
42 views

Can we throw away points to make a Holomorphic injection into a homeomorphism?

Let $U\subset\mathbb{C}$ and let $\varphi:U\to\mathbb{C}^n$ be a holomorphic injection. Is it true that there is a discrete (or better yet finite) set $Z\subset U$ such that $\varphi|_{U\backslash Z}$ ...
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0answers
60 views

Spherical derivative in several variables

This is more of a reference request, really. The spherical derivative of a holomorphic function (in one variable) $f$ is defined by $ f^\# := \frac{|f^{'}|}{1 + |f|^2}. $ Is there a corresponding `...
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1answer
28 views

real holomorphic field definition equivalence problem.

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 67 is this lemma: And when it comes time for the proof he sais that $(2)$ and $(3)$ are "tautological". I'm ...