# Tagged Questions

**0**

votes

**0**answers

108 views

### path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int ...

**0**

votes

**1**answer

119 views

### Ulam stability of homogeneous polynomials

Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable)
Is the following statement true:
For every $\epsilon$ there is a $\delta$ such that for ...

**4**

votes

**1**answer

381 views

### Hausdorff distance on algebraic curves

Introduction
Let $K$ be a nice closed domain in $\mathbb{R}^2$, for example the closed unit ball. Recall that the Hausdorff distance on the family $C(K)$ of nonempty compact subsets of $K$ is defined ...

**47**

votes

**9**answers

2k views

### Taking “Zooming in on a point of a graph” seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...

**9**

votes

**1**answer

1k views

### What is the wild fundamental group?

In the abstract of
Singularités irrégulières Correspondance et documents
Pierre Deligne, Bernard Malgrange, Jean-Pierre Ramis
Documents mathématiques 5 (2007), xii+188 pages (link)
there is ...

**6**

votes

**0**answers

298 views

### Riemann-Roch as an index theorem [closed]

I am sorry to make this a new question. I would have liked to leave a comment, but I suppose I don't have enough rep for that....
So, in the accepted answer to this question I don't understand why in ...

**2**

votes

**1**answer

1k views

### Puiseux series for roots of polynomials with smooth coefficients

If
$$p(x,y) = x^N + a_{N-1}(y)x^{N-1} + \ldots + a_0(y), \quad x,y \in \mathbf{C}$$
is a monic polynomial in $x$, and the coefficients $a_j$ are analytic functions of $y$, then the roots of $p$ ...

**1**

vote

**2**answers

355 views

### Finding regions where multi-variate polynomials are positive

Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P_j:\mathbb N^n\to \mathbb Z$}$\_{j=1...p}$, with $P_j \not\equiv 0$.
Is the following true:
There exists $n$ sets ...

**0**

votes

**1**answer

241 views

### the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n

we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the ...

**5**

votes

**3**answers

1k views

### Zariski open sets are dense in analytic topology

How does one show that if $U \subseteq \mathbb{C}^n$ is nonempty and Zariski open then $U$ is also dense in the analytic topology on $\mathbb{C}^n$?

**5**

votes

**3**answers

830 views

### Real-analytic manifolds in real-analytic sets

Let $U\subset \mathbb{R}^n$ be open, and let $f:U\to\mathbb{R}$ be real-analytic. We consider the zero set $Z:=f^{-1}(\{0\})$.
For a paper I am writing, I am looking for the best reference to the ...

**12**

votes

**7**answers

2k views

### Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. ...

**9**

votes

**3**answers

836 views

### level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $P$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube ...

**7**

votes

**1**answer

368 views

### Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...