0
votes
1answer
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
1answer
379 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 ...
46
votes
9answers
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
1answer
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
0answers
296 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
1answer
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
2answers
353 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
1answer
240 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
3answers
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
3answers
828 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
7answers
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
3answers
827 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
1answer
364 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 ...