Tagged Questions
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
-2
votes
1answer
149 views
Degree of a rational function [on hold]
I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
1
vote
1answer
28 views
About convex combinations of real-stable multivariable complex polynomials
Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
8
votes
2answers
239 views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...
2
votes
0answers
83 views
Bound on the sum of arguments
Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...
1
vote
0answers
128 views
About real roots of complex multivariable polynomials
Let $f(z,w_1,w_2,..,w_n)$ be a multivarible complex polynomial mapping $\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ and it has all real coefficients. Assume that this is "real-stable" i.e it has no roots ...
5
votes
1answer
358 views
Structure of the automorphism group of a Riemann surface
I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
16
votes
1answer
458 views
+50
Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?
There is a 50 point bounty on this question.
Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: ...
0
votes
0answers
73 views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
2
votes
0answers
79 views
Real-rooted polynomials and higher rank matrices
For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...
2
votes
1answer
223 views
Triviality of holomorphic vector bundles over contractible Stein manifolds
If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
-1
votes
0answers
49 views
Other Ways for Riemann Zeta Analytic Continuation [migrated]
A well-known way for analytic continuing riemann zeta function is using from the functional equation between $\zeta$, $\theta$ and $\Gamma$ function. but I know that there is or there are other ways ...
1
vote
0answers
49 views
Define an entire function with zeros in a given set [closed]
how to define an entire function that it's zeros are from a given set. for example, define an entire function that it's zeros are prime numbers on real axis.
0
votes
0answers
134 views
Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?
For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
1
vote
1answer
80 views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain ...
0
votes
0answers
45 views
Maximum modulus principle and restriction
Suppose I have a complex multivariate polynomial $f\in \mathbb{C}[z_1,\ldots,z_n]$ and a connected, bounded open set $U$. The maximum modulus principle says $f$ achieves its maximum value on the ...
5
votes
1answer
165 views
Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?
Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...
1
vote
1answer
115 views
How to find isothermal coordinates equivalent to circles in far limit?
I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...
3
votes
0answers
89 views
Implicit function theorem for singularities
I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
5
votes
1answer
215 views
Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?
Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f_n$ is a sequence of holomorphic functions converging pointwise to 0 on ...
0
votes
0answers
20 views
A bound in Schramm-Loewner evolutions
Recently, I read a paper in SLE that claims the following (which I do not really understand why):
Recall the differentiability estimate which says that $$|g_K(z) - z- \frac{\text{hcap}(K)}{z}| ...
3
votes
0answers
126 views
A difficult integral which the Risch algorithm shows is not elementary
For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
...
2
votes
0answers
103 views
Does this symmetrization operator have a name? Any theory?
Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define
$$f_{\mathrm{symm}}(x_1,\ldots,x_n) =
2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1}
...
0
votes
1answer
89 views
holomorphic continuation
consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.
This function can be continued holomorphically for all complex numbers with ...
1
vote
1answer
130 views
Sequence of smooth maps converging to the identity [closed]
Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
1
vote
1answer
67 views
Finding Laurent Series of a function [closed]
I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
0
votes
0answers
103 views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...
10
votes
1answer
208 views
Is there a solvable point on any variety over the field of complex rational functions?
Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...
2
votes
1answer
121 views
Flatness of a morphism of complex analytic spaces
Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...
3
votes
1answer
147 views
A weak analytic version of the valuative criterion of properness
EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...
1
vote
0answers
74 views
Possible generalizations of Hadamard's three line lemma
Let $f$ be an analytic function on a sector
$$
S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\}
$$
with opening angle $\gamma$ at the origin. Suppose $f$ is ...
2
votes
1answer
144 views
polynomial inequality in complex variable (generalized)
Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number.
What happens if ...
0
votes
1answer
154 views
existence of positive curved line bundles on a compact Riemann surface
Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
0
votes
0answers
50 views
Analytic functions space on Riemann surface
I have some questions about the analytic function space on Riemann surface and distinguished varieties:
Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...
7
votes
1answer
169 views
Analytic diffeomorphisms of the circle from complex domains
Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to ...
1
vote
0answers
171 views
Estimating the kernel of Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for complex $z$
Let $f(z,a)$ be an analytic function on $C^+=\{\Re z>0\}$ for each fixed $a>0$, and we have the following (weaker) estimates
$$
|f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
2
votes
0answers
72 views
Is logarithmic convexity of the heat kernel with complex time a general fact?
Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
7
votes
0answers
140 views
Complex structures on $\Bbb R^4$
Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...
2
votes
1answer
121 views
Complex function for mapping a circle to a superellipse
I was wondering if anyone knows an analytic complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated!
Thanks,
Kayvan
3
votes
0answers
97 views
Dimension of certain polynomial spaces
Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...
4
votes
2answers
239 views
Is there any elementary proof of No wandering domain for polynomials
It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...
0
votes
0answers
44 views
on the existence of holomorphic coordinates under bounded curvature
Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...
2
votes
0answers
69 views
Uniqueness of an embedding theorem for Real differential fields
I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...
1
vote
1answer
64 views
An optimization problem in complex space
Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...
7
votes
1answer
148 views
Conformal map of polygon with circle segments
I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a ...
13
votes
3answers
394 views
Entire function bounded at every line
I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
0
votes
0answers
65 views
kostant partition function vs Haar measure
I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant:
$$ \Delta(\theta) = \prod_{i< j} ...
4
votes
2answers
124 views
Bound areas of disks with respect to a quadratic differential
Let $q = q(z)\,dz^2$ be a quadratic differential on the unit disk $D(1)$, normalized so that $\int_{D(1)}|q| = 1$. I have a fairly convoluted argument that for any smaller disk $D(r)$ with $r < 1$, ...
4
votes
1answer
108 views
Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
8
votes
0answers
90 views
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 ...
1
vote
1answer
208 views
Residues and values of Riemann Zeta function at some points
I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...
