Questions tagged [cv.complex-variables]

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

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"Three great cocycles" in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
Kostya_I's user avatar
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28 votes
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Number of real roots of a polynomial

Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
Aleksei Kulikov's user avatar
23 votes
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1k views

Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by $$\gamma(K) := \sup |f'(\infty)|$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
Malik Younsi's user avatar
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Cartan–Oka vanishing in one variable without $\overline{\partial}$?

This is a literature question, about possible proofs of some very basic results in complex analysis. Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
Peter Scholze's user avatar
20 votes
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658 views

Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
Roland Bacher's user avatar
16 votes
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517 views

Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
Wolfgang's user avatar
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What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = \...
Greg Zitelli's user avatar
14 votes
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Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: $$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
mohammad-83's user avatar
13 votes
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167 views

Proofs of the valence formula that avoid tricky contours?

$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...
Terry Tao's user avatar
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Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&...
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Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
Gjergji Zaimi's user avatar
13 votes
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Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of $1$ complex variable, one ...
Qfwfq's user avatar
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12 votes
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Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$ where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...
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How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
Pedro's user avatar
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Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
Lasse Rempe's user avatar
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Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? I ...
Allen Knutson's user avatar
11 votes
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285 views

More identities for the Lambert $W$ function

In view of Iosif Pinelis's question about $\sum_{k\in\Bbb Z}1/(1+W_k(x))$, I played a little with $S(x,j)=\sum_{k\in\Bbb Z}1/W_k(x)^j$. It seems that if $x$ is a rational number with small denominator,...
Henri Cohen's user avatar
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11 votes
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Holomorphically convex manifolds and Bergman complete manifolds

Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
diverietti's user avatar
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11 votes
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Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
Kevin Smith's user avatar
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11 votes
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Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^...
diverietti's user avatar
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Adeles of Holomorphic Functions

In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
David Corwin's user avatar
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566 views

Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
Yuli Nazarov's user avatar
10 votes
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186 views

Local cohomology and residues of rational functions at 0 and $\infty$

Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where $s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...
Richard Stanley's user avatar
10 votes
0 answers
600 views

“Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble My question is similar to an earlier MathOverflow question: “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
Nike Dattani's user avatar
10 votes
0 answers
300 views

the (non-existent) group of conformal transformations

In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...
André Henriques's user avatar
10 votes
0 answers
349 views

Riemann–Hilbert-type problem

Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides of $P$ going in the counterclockwise order. We are ...
Misha's user avatar
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Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
Torsten Schoeneberg's user avatar
10 votes
0 answers
333 views

Analytic space not embeddable in any complex manifold

I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact). I ...
YangMills's user avatar
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10 votes
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196 views

Projective tensor squares of uniform algebras

In discussion with a colleague recently (Jan 2017), $\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$ I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
Yemon Choi's user avatar
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10 votes
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811 views

Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ...
Pascal Maillard's user avatar
10 votes
0 answers
527 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
Zen Harper's user avatar
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9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
pie's user avatar
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9 votes
0 answers
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From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis

I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define $$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
Vincent Granville's user avatar
9 votes
0 answers
318 views

When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
Wane's user avatar
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9 votes
0 answers
203 views

A geometric characterization of quasicircles

I'm reading an article by complex analysists. A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...
sharpe's user avatar
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9 votes
0 answers
260 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
Kevin Smith's user avatar
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9 votes
0 answers
458 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
Damian Rössler's user avatar
8 votes
0 answers
320 views

Who introduced the notion of ringed spaces?

My question is very concise, please forgive it. Who introduced the concept of ringed space? My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
user234212323's user avatar
8 votes
0 answers
241 views

Switching the order of a summation and replacing a series by its analytical continuation

Background A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
Caleb Briggs's user avatar
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8 votes
0 answers
168 views

Roots of a family of polynomials forming shapes

Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$. Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$. The roots of $F_n$ seems to form "shapes&...
LeechLattice's user avatar
  • 9,411
8 votes
0 answers
308 views

Is an entire function $\mathbb{C}^n\to\mathbb{C}$ a composition of polynomials, univariate entire functions and integrals?

Let $S$ be a set of entire functions $\mathbb{C}^n\to\mathbb{C}$. To enlarge it we can take polynomial combinations of its elements, compose them with entire functions $\mathbb{C}\to\mathbb{C}$ and ...
user avatar
8 votes
0 answers
284 views

On the remainder of a power series evaluated on the boundary of its convergence disk

Background This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
Daniele Tampieri's user avatar
8 votes
0 answers
304 views

The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures

Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
MCS's user avatar
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8 votes
0 answers
308 views

Singularities of a morphism from a smooth projective variety to an abelian variety

Let $f: X\to A$ be a (flat) morphism from a smooth complex projective variety $X$ to an abelian variety $A$. Consider the following natural diagram: $$T^*X\overset{df}{\longleftarrow}X\times H^0(A, \...
Feng Hao's user avatar
  • 1,071
8 votes
0 answers
251 views

Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
WeakMath's user avatar
8 votes
0 answers
209 views

Dense Stein subset in complex manifold

Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor. ...
D.Namrebod's user avatar
8 votes
0 answers
167 views

Padé Approximants of Power Series with Natural Boundaries

Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
MCS's user avatar
  • 1,256
8 votes
0 answers
275 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
asv's user avatar
  • 21.1k
8 votes
0 answers
104 views

What is known about the following series?

For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$. For $s\in\mathbb C$ let $$ D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
user avatar
8 votes
0 answers
175 views

Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let $X$ be a smooth separated ...
Tom Hawes's user avatar

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