# Tagged Questions

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: http://en.m....

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### Versal deformation of $x^3+y^3$

I am trying to compute fundamental group of complement to discriminant hypersurface of $f=x^3+y^3$ singularity via Zarisski-van Kampen theorem. So, I need a versal deformation of singularity to ...
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### Composition with polynomial/ same type of singularity

Let $f\in O(D_1(0){}-\{0\})$ and $p$ a non constant polynomial. Then $f$ and $p(f)$ have the same type of singularity at $z_o=0$. I think its fairtly easy to Show that if $f$ has a singularity ...
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### How to resolve the singularity of $xy+z^4=0$?

This singularity can not be resolved by one time blow-up. I don't know how to blow up the singularity of the "variety" obtained by the first blow-up, in other words, I am confused with how to do the ...
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### Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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### How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups

In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the ...
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### Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
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### Compute the genus of a curve with a flex point

The genus of a smooth plane curve is $g=\frac{(d-1)(d-2)}{2}$ and I know that if the curve has $n$ nodes the genus decreases by $n$. What happens if the curve has singular (non ordinary) points? In ...
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### On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
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### Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of poles,...
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### Local ring in singularity theory and algebraic geometry

In the book Singularity theory 1 by Arnold et al., page 13, there is a definition of a local algebra of singularity, which I tried and failed to rephrase in the language of algebraic geometry. It is ...
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### How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
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### Why must a meromorphic function, bounded near infinity, have the same number of poles and zeros?

Writing down some easy rational functions to check this, I don't see why this must be the case. Although if the function had 3 simple zeros and 2 simple poles its rational form would be in the form ...
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### How to numerically solve equation with regular singularity problem?

Good day, Here is my equation $$(x-1)^2 (x-h)^2q_1(x,\Omega)y''(x)+(x-1)(x-h)q_2(x,\Omega)y'(x)+q_3(x,\Omega)y(x)=0$$ ( with $q_1(x,\Omega)$[polynomial function 38 degree], $q_2(x,\Omega)$ [...
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### Finding the order of pole of $f(z)=\frac{\sin z}{z-\pi}$

The problem is Kreyszig 10ed international edition : 16.2 #9. What is the order of the pole at $z=\pi$ of the function $f(z)$ below? $$f(z)=\frac{\sin z}{z-\pi}$$ I thought that it will be a simple ...
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### Identification of pole

I would like to know whether the following statement is true or not If $f$ is an analytic function satisfied $|f(z)|\to\infty$ when $z\to z_0$, then $f$ is a pole at $z=z_0$ I would say the ...
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### Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
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### Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
The integral is $$\int_0^{\infty}\frac {1}{\sqrt{x}(1+x^2)}dx$$ which is to be evaluated by contour integration. So, the integrand clearly has simple poles at $+/- i$. But what kind of pole ...