**3**

votes

**1**answer

127 views

### Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows:
Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$
such that ...

**8**

votes

**2**answers

162 views

### Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ :
$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$
The discriminant of this polynomial in $r$ is the following expression
(obtained using ...

**6**

votes

**1**answer

205 views

### cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial
and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...

**3**

votes

**2**answers

258 views

### Approximation of curves

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...

**3**

votes

**1**answer

131 views

### Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...

**6**

votes

**0**answers

155 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

**6**

votes

**0**answers

65 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**12**

votes

**0**answers

245 views

### real algebraic geometry software?

Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...

**7**

votes

**3**answers

465 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 $X$ ...

**5**

votes

**0**answers

110 views

### moduli in real/semi algebraic geometry

Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry?
The sort of thing I am imagining is an object in a category of semischemes:
Ordinary schemes ...

**11**

votes

**1**answer

676 views

### What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.
Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...

**1**

vote

**0**answers

95 views

### Representing quasianalytic functions in several variables

For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...

**62**

votes

**8**answers

10k views

### How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...

**3**

votes

**1**answer

107 views

### Compactness of a semi algebraic set

Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?

**5**

votes

**1**answer

195 views

### Simple criterion to verify that the real zeros are an irreducible algebraic set

Given an irreducible polynomial $p$, its set of real zeros might be a reducible algebraic set. For example: $p=(x^2-1)^2 +(y^2-1)^2$.
Is there a simple sufficient condition on $p$ so that its real ...

**1**

vote

**1**answer

164 views

### Homogenous polynomial inequalities to prove positivity?

If I have $n$ variables $x_1, \cdots, x_n$, and a set $S$ of inequalities of the form $p(x_1, \cdots, x_n) > 0$ where $p$ is a homogenous real polynomial, is it true that I need $|S| \geq n$ to ...

**1**

vote

**0**answers

45 views

### Components of Intersection of Ellipsoids

Let $\Sigma$ be an $n-1$ dimensional ellipsoid in $\textbf{R}^{n}$ and $S$ the unit sphere.
I would like to understand the connected components $C$ of the intersection of $\Sigma$ and $S$.
In my ...

**5**

votes

**1**answer

209 views

### Are there general position results in singular algebraic sets?

Let $X$ be a real algebraic set, and let $Y \subset X$ be its singular set. In this question I'll focus on the analytic topology, so we can just imagine that $X$ is the zero set, in $\mathbb{R}^n$, ...

**4**

votes

**1**answer

276 views

### The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...

**2**

votes

**1**answer

172 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 ...

**2**

votes

**3**answers

183 views

### How to tell if a second-order curve goes below the $x$ axis?

Suppose we have a second-order curve in general form:
(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.
I'd like to know if there is a simple condition that ensures that the curve ...

**6**

votes

**0**answers

279 views

### Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case.
I call a partition ...

**0**

votes

**1**answer

139 views

### Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):
Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space
of ...

**6**

votes

**2**answers

375 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 ...

**4**

votes

**1**answer

582 views

### Exact arithmetic for real algebraic numbers

There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...

**2**

votes

**3**answers

598 views

### Solving a system of algebraic equations

I am trying to find complex solutions with positive real part $\{t_j \;|\;{\rm Re}\;t_j>0,
j = 1, 2, 3, \dots, n\}$ of the system of equations
$$0 = 1 + \sum_j \left(t_j^{2l+1} + ...

**1**

vote

**1**answer

87 views

### Decidability of the generated order

is the question whether a polynomial is non-negative on some semi-algebraic set (equivalently, is it in the cone denerated by some polynomials in the field of rational functions) known to be ...

**0**

votes

**1**answer

157 views

### Nonsingular zeroes are algebraic?

I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields):
Suppose $Q\in L^n$, $f_1, ...

**5**

votes

**0**answers

181 views

### Differential operators that preserve real-rootedness

Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of ...

**5**

votes

**1**answer

299 views

### Deciding the convexity of semialgebraic sets

Given a basic closed semialgebraic set, $S \subset \mathbb{R}^n$, defined by
$S = \{ x \in \mathbb{R}^n \mid g_1 (x) \geq 0 \land \dots \land g_m (x) \geq 0\}$
where $m \in \mathbb{N}$ and $g_1, ...

**4**

votes

**2**answers

497 views

### Real vs complex surfaces

Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these.
In particular I'm trying ...

**4**

votes

**1**answer

484 views

### Morley's Theorem and real algebraic geometry

Consider the following attempt at a ``thought-free'' proof of Morley's
Theorem.
Let $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ denote three vertices of
a generic triangle.
Let $(a_1,b_1)$, $(a_2,b_2)$ ...

**4**

votes

**1**answer

408 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 ...

**2**

votes

**1**answer

135 views

### Ultrafilters on the set of semialgebraic subsets of R^2

Let $R$ be a real closed field and $f: R \to R$ a map. Then let $\textrm{F}(f)$ be the set of semialgebraic subsets of $R^2$, which contain $(t,f(t))$ for all $0< t< \epsilon$ for some $\epsilon ...

**10**

votes

**1**answer

623 views

### Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)?

Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the ...

**1**

vote

**0**answers

211 views

### Trigonometric semialgebraic conditions for two floors to be unequal [edited]

EDIT: Since posting my original question I have simplified the problem to finding a sufficient condition for $\lfloor \frac{a}{\phi}\rfloor \neq \lfloor \frac{b}{\phi}\rfloor$ which is trigonometric ...

**3**

votes

**4**answers

508 views

### Computation of the Euler characteristic of a specific real variety

I think computation of the Euler characteristic of a real variety is not a problem in theory.
There are some nice papers like J.W. Bruce, Euler characteristics of real varieties.
But suppose we ...

**0**

votes

**0**answers

212 views

### Ideal of real points in $\mathbb{C}[x_0,x_1,\dotsc,x_k]$

I fell a little uncomfortable with this real stuff. The question here is more general, but we can suppose $\mathbb{K}=\mathbb{R}$.
Take a set of (distinct) points in $\mathbb{P}^n$, the complex ...

**0**

votes

**1**answer

366 views

### Degree of a real algebraic variety and regular morphisms

I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety ...

**1**

vote

**1**answer

389 views

### Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED]

Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively.
Let $G_1,\ldots, G_l$ be their ...

**4**

votes

**2**answers

556 views

### intersections of real algebraic sets (a bezout-type question)

This question arose when I was trying to understand the Guth-Katz paper on Erdos distance problem, namely, related to Proof of Lemma 2.9 in http://arxiv.org/abs/1011.4105. The proof of that lemma is ...

**3**

votes

**1**answer

478 views

### Quick algorithm for finding real solutions for a system polynomial equations

Would you please recommend a computer program that could give quick answer Yes/No for the question: does there exist a real solution of a given system of polynomial equations with integer ...

**8**

votes

**2**answers

441 views

### The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy type of the semi-algebraic set defined by
$$ P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r . $$
Is there a ...

**2**

votes

**0**answers

273 views

### Integrating over the real points of an algebraic variety

This question is motivated by my earlier question.
Suppose I have some algebraic variety $X/\mathbb R$ of dimension $n$, and suppose that $X(\mathbb R)$ is compact. Now for any element of ...

**6**

votes

**2**answers

680 views

### Maximal number of connected components of complement to an affine plane real algebraic curve

Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?

**7**

votes

**1**answer

655 views

### (Real) algebraic geometry for (real) trigonometric polynomials?

Has somebody developed a comprehensive theory of the algebraic structure of trigonometric polynomials in several variables? If yes, where?
Background:
By a (real) trigonometric polynomial in ...

**2**

votes

**0**answers

302 views

### Maximal compact subgroup of a real semisimple Lie group of “quasi-adjoint” type.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and $G$ the adjoint group of $\mathfrak{g}$.
Let $\sigma:\mathfrak{g}\rightarrow \mathfrak{g}$ be a complex conjugation and ...

**4**

votes

**1**answer

252 views

### Hypersurfaces without real points

Let $n, d$ be positive integers. I am interested in the open subset
$\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to ...

**13**

votes

**5**answers

2k views

### Cohomology of Structure Sheaves: Algebraic, Constructible and more

I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...

**1**

vote

**3**answers

813 views

### Isolated zero set of real polynomial in two variables

The polynomial $x^2+y^2$ has an isolated zero at the origin. And so do powers $(x^2+y^2)^n$ of this polynomial. I'm wondering if this is a special property of these real polynomials.
Here's the ...