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Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

4
votes
1answer
175 views

$L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
6
votes
1answer
242 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
4
votes
1answer
149 views

Is there a converse to Ikehara's Tauberian theorem for Dirichlet series?

Ikehara's Tauberian theorem for Dirichlet series states that if $$F(s)=\sum_{1}^{\infty}\frac{f(n)}{n^s}$$ with $f(n)\geq 0$ is such that $$F(s)=\frac{G(s)}{s-1}+H(s)$$ for $\sigma>1$ with $G,H$ ...
0
votes
0answers
85 views

What can be said about the level set of the real part of an analytic function?

I am cross-listing this question from math.stackexchange since I did not find a satisfactory answer there. This is my first time posting a question on MO, so if this is not the appropriate community I ...
15
votes
3answers
824 views

Is this lower bound for a norm of some complex matrices true?

Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
0
votes
0answers
58 views

Upper bound on the modulus of a power series and concentration inequalities for empirical processes

This is a research question I encountered when I as studying solutions of Lebesgue-Stieltjes integral equations. It is related to a new statistical method I am developing (which I cannot expose now) ...
3
votes
0answers
103 views

Asymptotic Expansion of Seiberg-Witten Differential?

Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by \begin{equation} \mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
1
vote
0answers
62 views

The Geometry of Jacobi Forms and their Asymptotic Expansions

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying $$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
6
votes
0answers
148 views

Reference request: normal form of k-differentials and flat surfaces at a puncture

Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
1
vote
0answers
37 views

Finitely connected planar domains and certain smooth maps with integrable Wirtinger derivative

Let $D$ be a finitely connected, bounded domain in $\mathbb C$. Suppose that the boundary of $D$ consists of finitely many, pairwise disjoint, piecewise $C^1$-Jordan curves. Does there exist an ...
3
votes
1answer
175 views

Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?

Nakajima & Yoshioka [1] showed that \begin{equation} F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
5
votes
1answer
101 views

Padé multipoint approximants of the exponential function

One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with $\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
4
votes
0answers
76 views

Extending a holomorphic map on diffeomorphic affine varieties

Suppose I have two smooth complex affine varieties $X$ and $Y$. Assume that they are each diffeomorphic to $\mathbb{R}^{2n}$ (where $n\geq 3$). Question: If there exists open dense subsets $U\...
1
vote
0answers
59 views

Dirichlet series with an abscissa of absolute convergence $\sigma_{0}$, analytic in $\sigma > \sigma_{0} - \delta$

Suppose that a Dirichlet series $f(s)$ has the abscissa of absolute convergence $\sigma_{0}$ and is analytic in $\sigma > \sigma_{0} - \delta$ for some $\delta > 0$. For $\sigma > \sigma_{0}$,...
0
votes
0answers
53 views

Schlesinger theorem

I am trying to understand a proof of Schlesinger theorem but I don't succeed. Schlesinger theorem: Let $Y'=A(z)Y$ be a differential equation with coefficients of $A$ in $k=\mathbb{C}(z)$. If this ...
4
votes
3answers
298 views

Non combinatorial random matrix theory

I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, ...
1
vote
1answer
54 views

Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
0
votes
1answer
118 views

Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

Is anything useful known about the function defined by \[ f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ? \] Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...
31
votes
2answers
3k views

Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
3
votes
1answer
195 views

Complex integral

We would like to compute (or bound) the following complex integral: $$\int\limits_0^{\infty}\left|\int\limits_{-\infty}^{\infty}\frac{e^{its}}{e^s-\lambda}\,ds\right|\,dt$$ where $\lambda \notin S_{\...
2
votes
1answer
130 views

asymptotic behavior of a singular integral

What is the asymptotic behavior of the integral $$\int\int_{\mathbb C}\frac{f(z, \bar z)}{(z-a)(\bar z- \bar a_0)}|dz|^2$$ as $a\to a_0$? The function $f$ is from $C^\infty_0({\mathbb C})$ with ...
2
votes
2answers
175 views

An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$

For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
1
vote
0answers
76 views

An equality of inner products of holomorphic curves

The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The ...
3
votes
0answers
75 views

Hunting for holomorphic functions part 2: this time with non-local boundary condition

This is a follow-up to Looking for holomorphic function on a sector with specified boundary behavior. There I was looking for a holomorphic function on a sector with real boundary condition on one ...
2
votes
0answers
77 views

Super global dimension

Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$. Here $id(X)$ stands for the injective ...
7
votes
1answer
262 views

lower bound for absolute value of a hypergeometric function

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$. It appears that $\left| _{2}F_{1}(a,a-b;2a;1-...
7
votes
1answer
243 views

Looking for holomorphic function on a sector with specified boundary behavior

Fix $h \in (0,\pi/2)$. I am trying to explicitly exhibit a holomorphic function $f\colon \Sigma \to \mathbb{C}$, where $\Sigma$ is the punctured sector $$\Sigma := \{z \in \mathbb{C} \:|\: z\neq 0, 0\...
6
votes
1answer
506 views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
1
vote
0answers
48 views

Iterates of analytic functions and fixed points

My question grew from the following $1-$dimensional result. Let $A,B\subset \mathbb{C}$ be open connected neighbourhoods of $0\in\mathbb{C}$ and $U:A\times B\to \mathbb{C}$ be an analytic function. ...
3
votes
2answers
94 views

Decay of bandlimited function

Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have. I read on "G. ...
2
votes
0answers
117 views

Distribution of random hyperplanes in projective spaces

Let $X\subset \mathbb{CP}^{N-1}$ be a smooth subvariety of dimension $n$. Assume that $X$ is not contained in a hyperplane of $\mathbb{CP}^{N-1}$. Let $\mu$ be a smooth probability measure on $X$. ...
0
votes
0answers
127 views

On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...
4
votes
1answer
153 views

Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function $$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, ...
0
votes
0answers
54 views

Closed Form or Special Function Expression for Basic Looking Geometric-Like Sum

Is there any known closed form, involving any commonly used special function, of sums of the type $$\sum_{k=0}^{n-1} a^{k^m}$$ for m = 2, 3, ... It can be generalized to $m$ being a general complex ...
4
votes
0answers
106 views

Nekrasov Partition function and the leading term of Prepotential

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf. In (4.25) the author expressed the partition function ...
0
votes
0answers
141 views

Section of a holomorphic line bundle with given differential at a zero

Let $X$ be a compact Kähler manifold of dimension $n$ with a given Kähler metric $\omega$. Let $L$ be a hermitian holomorphic line bundle on $X$ whose metric is positive. Let $x_0\in X$. I would ...
5
votes
3answers
245 views

fixed points of quadratic iteration

Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period ...
2
votes
1answer
79 views

Analytic sections of a GIT quotient lying in the Kempf-Ness set

I would like to understand whether the following is true. Given a complex reductive group $G$ (for me $G = \operatorname{PGL}_n(\mathbb{C})$) acting on a vector space $V$ (for me $V = M_n(\mathbb{C})^{...
1
vote
1answer
100 views

Bound of the measure of the support of a set of divisors in a fixed linear system

Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure ...
5
votes
0answers
88 views

Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?

Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
3
votes
1answer
122 views

Trees and Shabat polynomials

Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no ...
1
vote
0answers
94 views

Asymptotics of an integral by two methods

This was asked in MSE, here, but the answer was not satisfactory. I want to compute the asymptotic behavior of the integral $$ f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$ when $K$ is large ...
1
vote
0answers
107 views

Differentials on tori realised as double of annuli

In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus. In short, the torus is realised ...
0
votes
0answers
101 views

$\det(I-K(z)+\varepsilon(z,x)) $ versus $\det(I-K(z))$

First let me ask the general question that might interest others dealing with determinantal formulas. We are trying to compare the following two quantities $$C_{\varepsilon} := \oint \det(I-K(z)+\...
2
votes
0answers
78 views

Euler characteristic of an exhaustion of compacts of a surface

Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$ ...
5
votes
0answers
143 views

The Krzyż Conjecture

What is the state of the Krzyż Conjecture? It states for that for all $f:\mathbb{D}\to \bar{\mathbb{D}}$ holomorphic and non-vanishing, the coefficients $a_n$ in the power series of $f$ are at most $2/...
1
vote
0answers
117 views

Path components of complex analytic spaces [closed]

Let $(X,\mathcal{O}_X)$ be a complex analytic space. Are the path components of $X$ the same as the connected components?
4
votes
1answer
137 views

An estimate on deviation of two smooth tangent $J$-holomorphic curves

Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to ...
1
vote
1answer
56 views

Is the distribution of the real part of product of two independent complex variates exponential?

Trying to find the pdf of the real part x of the product $z_1z_2$ of two uncorrelated complex random Gaussian variates . The pdf of the modulus $r \equiv |z_1z_2|$ is known $ f_r(r)=rK_0(r)$ from ...
4
votes
0answers
117 views

Inclusion of Hardy spaces

It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality. It is also known that for $p>1$ it holds that $L^p(\mathbb R)...