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).
2
votes
0answers
22 views
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 ...
1
vote
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
votes
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 ...
2
votes
0answers
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 ...
1
vote
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
$$f:U\rightarrow\...
1
vote
0answers
17 views
Is there an extension of the concept of Stokes lines/the Stokes phenomenon to 2 complex variables?
For one complex variable, normally we refer to the stokes(/antistokes) lines as being lines that divide the input space according to asymptotic behaviour.
I hope we can extend this idea to "stokes ...
2
votes
0answers
18 views
references to boundary behaviour d-bar problem solutions
is there any reference around the subject of boundary behaviour of solutions of $\bar\partial$ problem $\bar\partial u=f$, when $f$ as slow growth at the boundary of a pseudo-convex domain $\Omega$ in ...
1
vote
2answers
62 views
Definition of divisor associated to a meromorphic function
Let $X$ be a complex manifold and let $f$ be a meromorphic function defined on it. Let $Div(X)$ be the group of locally finite sum of analytic irreducible hypersurfaxces of $X$.
One would like to ...
0
votes
1answer
38 views
Does there exist a function $f:\mathbb{C}^2\to \mathbb{C}$ holomorphic aside from at singularities with isolated singularities?
All the examples I've been able to come up with are either non-holomorphic or fail the condition. For example, with the 1D case we can take $$f(z) = \frac{1}{1+z^2}$$ With singularities at $\pm i$. If ...
0
votes
2answers
23 views
Taylor series of complex multivariate function
I am looking for a reference or some literature on Taylor series of complex multivariate functions. I found material for complex functions and material for multivariate functions, but not for both.
...
1
vote
0answers
71 views
Factorization of an analytic function in $\mathbb{C}^n$
Let $\Omega$ be an open set of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one.
If $...
1
vote
0answers
18 views
$\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ s.t. $Q_i$ is a prime ideal.
Prove that the ideal $\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ such that $Q_i$ is a prime ideal.
0
votes
0answers
21 views
$A\cap\Lambda$ polynomially convex in $\Lambda$
Let $A\Subset\Bbb C^n$ polynomially convex, that is its convex hull
$$
\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 $\...
0
votes
0answers
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 ...
1
vote
0answers
19 views
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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}):\...
1
vote
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 ...
2
votes
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 ...
3
votes
0answers
35 views
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 ...
1
vote
0answers
11 views
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 $\...
0
votes
0answers
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 ...
0
votes
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.
2
votes
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$ ...
1
vote
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 + ...
2
votes
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 ...
0
votes
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?
11
votes
1answer
181 views
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 ...
0
votes
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 $z$ ...
1
vote
0answers
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 $(x,y)\to x$....
3
votes
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):...
1
vote
0answers
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\...
0
votes
0answers
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}|...
0
votes
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:
$$x = \frac {- \sqrt2}{3(3^{1/4})}, y = \frac{-2}{\sqrt3}$$
0
votes
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 ...
2
votes
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
...
0
votes
0answers
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}\...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
1
vote
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 ?
0
votes
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-...
0
votes
0answers
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 $(...
1
vote
0answers
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 \...
0
votes
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 ...
1
vote
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 $$\...
0
votes
0answers
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 ...
0
votes
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 $\varphi|_{U\backslash Z}$ ...
0
votes
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 `...
1
vote
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 ...
