0
votes
0answers
112 views

When is there a polynomial transformation? [on hold]

First part: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial ( that ...
5
votes
2answers
872 views

Is it meaningful to work on convergencies, integration, etc. on the Zariski topology?

Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...
7
votes
3answers
428 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
2answers
266 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
0answers
85 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 ...
4
votes
2answers
594 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
0answers
152 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 $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
2
votes
1answer
160 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
1answer
118 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
2answers
348 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
0answers
151 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 ...
4
votes
2answers
366 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 ...
47
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 ...
4
votes
0answers
381 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. ...