# Tagged Questions

**7**

votes

**3**answers

407 views

### Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$).
We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...

**1**

vote

**2**answers

242 views

### Bounds on the largest root of a polynomial

Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...

**0**

votes

**0**answers

79 views

### Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...

**3**

votes

**2**answers

537 views

### Do proper Zariski closed sets of algebraic sets have measure zero

This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...

**1**

vote

**0**answers

142 views

### When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...

**0**

votes

**0**answers

32 views

### Last Point of Exit from the 2-D Positive Quadrant

I define the functions $f_i(\mathbf{u}),i=1,2$ and $g_i(\mathbf{u}),i=1,2$ where $\mathbf{u}\in\mathbb{C}^{N}$ is the unit-norm vector. Thus, this functions are defined over the unit norm $N-$sphere. ...

**2**

votes

**1**answer

152 views

### Volumes of families of semialgebraic sets

Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is ...

**1**

vote

**1**answer

114 views

### If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?

Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...

**6**

votes

**2**answers

342 views

### orderings of the field R((x, y))

I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...

**1**

vote

**0**answers

149 views

### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta ...

**3**

votes

**2**answers

350 views

### implicit function theorem for algebraic sets

We know by the standard Implicit Function Theorem that
If $f:\mathbb R^4\rightarrow\mathbb
> R^2$ is a polynomial (or in fact any
continuously differentiable function),
then there is a ...

**43**

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 ...

**4**

votes

**0**answers

370 views

### Cohomology of Real algebraic Varieities

I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...