The tag has no wiki summary.

learn more… | top users | synonyms

10
votes
1answer
497 views

Is there a classification of isolated essential singularities?

In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked: Do we not care about essential singularities to classify ...
6
votes
3answers
83 views

How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
5
votes
2answers
604 views

Singularities of $e^{z - \frac{1}{z}}$

I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
5
votes
1answer
101 views

Distribution with singularities.

I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty \psi^{(n)}(n)$ is a distribution which has singularities of infinite order. Here $\psi$ is a test function ...
5
votes
1answer
98 views

germ finitely determined

Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: ...
4
votes
3answers
906 views

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
4
votes
1answer
226 views

How to analyze the asymptotic behaviour of this integral function?

Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$ Without working out ...
4
votes
2answers
139 views

Do the polynomial germs generate all the ring of germs?

I'm trying to understand some equality that comes up in stability theory involving sets of germs and I think I need a result like the next one, so if anyone knows anything about this and helps me it ...
4
votes
1answer
93 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
4
votes
2answers
203 views

Singularities of $f(z)=z/\cos(z)$

Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not. I am a bit confused ...
4
votes
0answers
98 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
3
votes
2answers
49 views

Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
3
votes
2answers
329 views

Proving that a function has a removable singularity at infinity

I'm having trouble with the following exercise from Ahlfors' text (not homework) "If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that ...
3
votes
1answer
95 views

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$?

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$? If so, is it not meaningful to discuss its residue at $0$?
3
votes
1answer
74 views

Classify Singularities

So, I'm trying to classify the singular points of the following function: $$ f(z)=e^{\cot(\frac {1}{z})} $$ Obviously, when z is zero, the function tends to approach infinity, so that must be a ...
3
votes
1answer
49 views

Why is $P$ singular

This is from Shafarevich's 'Basic Algebraic Geometry 1': Let $P=(\alpha,\beta)\in X$, and suppose that the equation of $X$ is written in the form $f(x,y)=a(x-\alpha)+b(x-\beta)+g$, where $g$ is a ...
3
votes
1answer
72 views

Discrepancy non log canonical singularities

Suppose that $Y$ is a normal variety such that its canonical class $K_Y$ is $\mathbb{Q}$-Cartier. , and let $f:X \to Y$ be a resolution of the singularities of $Y$. Then $$ K_X= f^*(K_Y)+\sum_i ...
3
votes
0answers
39 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
3
votes
0answers
47 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
2
votes
2answers
293 views

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
2
votes
1answer
292 views

$z_0$ non-removable singularity of $f\Rightarrow z_0$ essential singularity of $\exp(f)$

Let $z_0$ be a non-removable isolated singularity of $f$. Show that $z_0$ is then an essential singularity of $\exp(f)$. Hello, unfortunately I do not know how to proof that. To my ...
2
votes
2answers
507 views

Contour integral $\int_{|z|=1}\exp(1/z)\sin(1/z)dz$

Evaluate the contour integral $$\int_{|z|=1}\exp(1/z)\sin(1/z)\,dz$$ along the circle $|z|=1$ counterclockwise once. The singularities are $\dfrac1{\pi k},k\in\mathbb{Z}$ plus the limit point $0$. So ...
2
votes
1answer
64 views

Do singularities always appear on all Riemann sheets?

Consider a function $$f(z) = \frac{\ln z}{z^2+1}.$$ Besides the branching point $z=0$, the function also has singularities at $z = \pm i$. This singularities should appear on all Riemann sheets. Is ...
2
votes
1answer
71 views

Local rings and classifying singularities

My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
2
votes
1answer
96 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
2
votes
1answer
62 views

Removable singularities for continuous functions

Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ ...
2
votes
1answer
39 views

Conditions for the number of degenerated fibers of a morphism to be finite.

I'm having trouble to find a theorem about this: Let's $F_z$ be a family of curves defined by $$F_z : F(x,y) + z = 0$$ where $F$ is an irreducible polynomial and $z \in \mathbb{C}$. My question is: ...
2
votes
1answer
51 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
2
votes
1answer
178 views

How to describe the singularities of a function

This is a question of an old exam of my complex analysis course and although I think I understand what a singularity is, I oftentimes have troubles 'finding and describing' them properly. I looked at ...
2
votes
1answer
3k views

Singular points of ODE

My friend and I have conflicting answers and since his phone is off, I can't get his full solution and I don't understand his argument. Consider this ODE $$(x+1)y''+\frac{1}{x}y' + (x+3)y= 0$$ ...
2
votes
0answers
75 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
2
votes
1answer
91 views

Removable singularities of a holomorphic function

So, I'm a little confused about removable singularities. Consider the function below: $$f(z)=\frac{1}{(1+z^2)^{2/3}}$$ Obviously, we have isolated singularities at the points $z = \pm i$. ...
2
votes
0answers
53 views

necessary and sufficient conditions for having an irreducible hyperplane section

Let $S \subset \mathbb{P}^3$ be a surface different to a cone. If $S$ is a smooth variety, then its generic hyperplane section is irreducible. I can imagine a similar result holds for a surface with ...
2
votes
0answers
53 views

Relation of two definitions of 'Singular Point'

In my lecture we defined: For $(a,b)=P\in V(f)$, $P$ is a singular point if $\operatorname{mult}(f,P)>1$, where $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
2
votes
0answers
55 views

The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
2
votes
1answer
82 views

Singularities in (Elementary) Real Algebraic Geometry

I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly ...
2
votes
1answer
61 views

Laurent series of an analytic function divided by $z$

This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$. A ...
2
votes
0answers
75 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
2
votes
0answers
42 views

I have an infinite solution to an ODE even though it has only a regular singular point

I have the ODE: $\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$ I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
2
votes
0answers
43 views

Algorithm for checking that a singularity is an ordinary double point

Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
2
votes
0answers
48 views

How do I check whether an orbifold admits deformations?

Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $n^\text{th}$ root of unity, admit smooth deformations. This is because ...
2
votes
2answers
174 views

The notion of a germ in singularity theory

I quote from my lecture: Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset ...
1
vote
2answers
145 views

Are the convergence radii circles of a Laurent-series always caused by isolated singularities?

Laurent series $$f(z) := \sum_{n=-\infty}^\infty a_n (z-c)^n$$ converge for $r<|z-c|<R$ where $$r = \limsup_{n\to\infty}|a_{-n}|^{\frac1n}, \\\frac1R = \limsup_{n\to\infty}|a_n|^{\frac1n}.$$ ...
1
vote
1answer
164 views

Prove that $f(z)$ can not be a polynomial

Suppose $f(z)$ and $g(z)$ are entire functions and that $f(z)$ is not constant. If $|f(z)| < |g(z)|$ for all $z \in \mathbb C, $ prove that $f(z)$ can not be a polynomial. I was thinking what I ...
1
vote
2answers
38 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
1
vote
1answer
514 views

Invertibility of a square matrix with zero diagonal elements and positive non-diagonal elements

$M$ is square and $$M(i,j)=0, i=j$$ $$M(i,j)>0, i\ne j$$ Is $M$ full-rank or invertible? Actually the $M$ I am studying has much stronger properties but I guess the simple conditions above might ...
1
vote
1answer
90 views

Isolated Singularities

Consider the following functions and determine which kind of singularities they have in $z_0$. If it is a removeable singularity, then calculate the limit; if it is a pole, then give the order ...
1
vote
1answer
98 views

Invertibility of matrix with each element equal to cofactor

I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
1
vote
2answers
50 views

Classifying singularity

Having trouble classifying a singularity... $f(z)=$$z^2-1\over z^6+2z^5+z^4$ with $z_0=0$ and $z_0=-1$ The $z_0=0$ is pretty simple, just need to put $z^4$ in evidence. But $z_0=-1$ I can't seem to ...
1
vote
1answer
52 views

coefficients for getting a smooth equation

I'm studying the equation $$ x_0q_0 + x_1q_1 + x_2q_2 = 0$$ where $q_i$ is a homogeneous polynomial of degree two in the variables $x_0,\ldots,x_5$. I would like to have some simple choices for the ...