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Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).

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Maximum number of connected components of a real affine curve

Harnack's curve theorem tells us that the maximum number of connected components of an algebraic curve of degree $d$ in the real projective plane is $1 + (d-1)(d-2)/2$ (and this bound is sharp). What ...
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35 views

Affine automorphisms of real affine varieties

Let $V \subset \mathbb{R}^d$ be a real affine variety. I'm hoping I will not butcher existing nomenclature too badly if I say that for the purposes of this question an affine automorphism of $V$ is an ...
10
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2answers
153 views

The set of polytopes with given $f$-vector

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
6
votes
0answers
80 views

another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$ In 1984 S.D.Berman, a Soviet mathematician, ...
2
votes
0answers
109 views

Chebyshev-like Problem for Plucker Coordinates

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$ Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $...
11
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0answers
187 views

Matrices that admit a power that is symmetric

We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
15
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1answer
473 views

How can I distinguish a genuine solution of polynomial equations from a numerical near miss?

Cross-posted from MSE, where this question was asked over a year ago with no answers. Suppose I have a large system of polynomial equations in a large number of real-valued variables. \begin{align} ...
16
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1answer
505 views

Counting real zeros of a polynomial

I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...
5
votes
1answer
75 views

(Euclidean) open orbit in an irreducible real algebraic set

Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
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votes
0answers
107 views

About the multiplicative group of p-adic complex

I was studying the multiplicative group of the $\mathbb{C}_p$. I'm interesed in the ring $\mathcal{O}_p$ of elements in $x\in\mathbb{C}_p$ such that $|x|_p\geq 1$. I have three questions. The first ...
3
votes
1answer
94 views

Atoric equation

I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that? ...
2
votes
1answer
83 views

Symmetric orthogonal matrices with constant diagonal entries

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\...
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votes
1answer
336 views

A question on symmetric matrices

$\newcommand{\R}{\mathbb{R}}$ The question is Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
3
votes
1answer
123 views

Solutions to a system of homogeneous equations (inequalities)

Let $f_1,\ldots,f_r \in \mathbb{R}[x_1,\ldots,x_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$. For which values of $r,n,d$ there exists a real ...
17
votes
1answer
458 views

An explicit reconstruction of a matrix from its minors

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ $\newcommand{\Cof}{\operatorname{cof}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
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0answers
131 views

Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \End(...
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1answer
168 views

Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?

$\newcommand{\End}{\operatorname{End}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define $H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...
5
votes
1answer
133 views

If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
2
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3answers
161 views

Algebraic inequalities on different means

If $a^2+b^2+c^2+d^2=1$ in which $a,b,c,d>0$, prove or disprove \begin{equation*} \begin{aligned} (a+b+c+d)^8&\geq 2^{12}abcd;\\ a+b+c+d+\frac{1}{2(abcd)^{1/4}}&\geq 3. \end{aligned} \end{...
12
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1answer
356 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
3
votes
0answers
39 views

Points of intersection of summand of sums of squares of real polynomials

$\newcommand\R{\mathbb R} \newcommand\Q{\mathbb Q} $I am thinking of something related to Blekhermans 2012 paper Nonnegative Polynomials and Sums of Squares (Journal of the AMS, 25, 2012, 617-635). ...
27
votes
1answer
583 views

$\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ? A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take ...
11
votes
1answer
247 views

First order decidability of limit of gradient flow?

Let $f: \mathbb{R}^n\to\mathbb{R}$ be a polynomial function, and let $p$ be a critical point. Consider the ascending manifold $A_p$ consisting of all points whose limit under the gradient flow of $f$ ...
4
votes
0answers
85 views

Cylindrical Decomposition vs Morse decomposition

Suppose I have a polynomial Morse function $f: \mathbb{R}^n \to \mathbb{R}$. Consider the ideal $I(\nabla f)$ generated by the partial derivatives $\partial_i f$, and assume that the real zero-set of ...
0
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0answers
62 views

A system of polynomial equations has exactly one positive real zero

Recently I consider the following system of polynomial equations: \begin{equation} \sum_{i=1}^m c_i(\boldsymbol{\alpha}_i-\boldsymbol{\beta}_r)\mathbf{x}^{\boldsymbol{\alpha}_i}-\sum_{j=1}^{r-1}d_j(\...
2
votes
0answers
27 views

Explicit formulas for polynomial invariants of cubic surfaces in Sylvester standard form

By a cubic surface $X_F$ we mean the zero locus of a homogeneous cubic polynomial $F(x,y,z,w)$. The group $\text{GL}_4$ acts on $X_F$ via substitution. The ring of polynomial invariants induced by ...
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1answer
108 views

a linear programming problem

Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give ...
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0answers
40 views

Question on the boundary of a image of a polynomial map

Let $\Phi_1(w_1, w_2, v_1, \ldots, v_n)$ and $\Phi_2(w_1, w_2, v_1, \ldots, v_n)$ be two polynomials in $n$ variables. Let $B = [a,a'] \times [b,b'] \subseteq \mathbb{R}^2$. For each fixed $\mathbf{v}$...
4
votes
1answer
122 views

Semialgebraic sets containing irrational power functions

Let $\alpha$ be an irrational number, and consider the set $A=\{(x,x^\alpha),x\ge 0\}\subseteq \mathbb{R}^2$, which is the graph of the function $f(x)=x^\alpha$. I'm trying to prove/disprove the ...
2
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0answers
100 views

Classification of $n$-dimensional Nash-submanifolds of $\mathbb{R}^n$

Let $M,N\subset \mathbb{R}^n$ be two open semi-algebraic subsets, and assume that $M$ and $N$ are $C^\infty$ diffeomorphic, i.e. isomorphic as smooth submanifolds of $\mathbb{R}^n$. Does this imply ...
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321 views

On a question of Coste & Roy from 1979

On page 44 of their 1979 paper, Topologies for real algebraic geometry, Coste & Roy define a structure sheaf on the real Zariski spectrum of a commutative ring (which can be regarded as the real ...
7
votes
1answer
199 views

Do convex closed semialgebraic hyperplane cross-sections imply semi-algebraicity?

Let $S\subset\mathbb{R}^n$, with $n\geq 3$, such that for any hyperplane $L$ one has $L\cap S$ closed, semialgebraic, and convex. Is it true that $S$ itself is semialgebraic? A colleague explained to ...
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0answers
302 views

Cohomology of a blow-up of a real algebraic variety

Let $X$ be a complex algebraic variety, $Z \subset X$ a closed subvariety, $\mathrm{Bl}_Z X$ the blow-up and $E$ the exceptional divisor. There is an isomorphism of cohomology groups $$ H^k(X(\mathbf ...
14
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1answer
288 views

Inequalities on elementary symmetric polynomials

I have recently come across the following result. Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
4
votes
1answer
87 views

Specific quaternary quartic that is positive semi-definite but not sum of squares

Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...
4
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1answer
171 views

Estimating the volume of a region bounded by polynomial inequalities

Let $Q(x,y,z)$ be a geometrically irreducible quadratic form in $x,y,z$ with real coefficients, such that $z^2$ appears with non-zero coefficient. Define the region $\mathcal{R}(X)$ by $$\...
2
votes
0answers
48 views

Unboundedness of number of solutions of intersection of bivariate polynomial with graph of function from an o-minimal structure

I am trying to understand a construction sketched in the paper by Gwozdziewicz, Kurdyka and Parusinski in the Proceedings of the AMS 1999 (paper here) and I'd like to request some help. The ...
2
votes
0answers
69 views

Sheaves of functions on open semi-algebraic sets

Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called (1) ...
5
votes
1answer
217 views

Are continuous rational functions arc-analytic?

Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...
3
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0answers
91 views

Question about Nash functions

I am reading Kollár's recent survey on Nash's work in algebraic geometry. I am trying to understand why the retraction $\pi:U_M\to M$ introduced in Discussion 7 is a Nash map. Kollár applies Claim 8.4 ...
5
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0answers
69 views

Smoothness for real closed spaces

Is there a notion of smoothness for maps of real closed spaces in the sense of Schwartz? [1] Ideally it would have the following properties: Every smooth map of real closed spaces is locally of the ...
7
votes
1answer
233 views

Does generic projection into $\mathbb{R}^3$ preserve real-algebraic-curve-ness?

I'm interested in the topological properties of certain real algebraic curves in high-dimensional spaces. I want to visualize these curves (say, like this), and so I'm pursuing dimensionality ...
4
votes
2answers
769 views

Every real variety contains non-singular points

I am looking for a relatively "elementary" proof that every variety in ${\mathbb R}^n$ contains at least one non-singular point. So far I only have such a proof for the case of hypersurfaces. ...
3
votes
2answers
209 views

How to find (or numerically find) a solution to a system of 10 equations with 14 variables?

I am trying to construct a counterexample in $\mathbb{R}^4$. There are 10 relations, each of which is given as the vanishing set of a determinant. There are 14 variables in total. Is there a way to ...
3
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1answer
109 views

Connectedness properties of real algebraic set

Let $p:\mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial with non-empty zero set $S$. Is it true that for any $x,y$ in the same connected component $C$ of $S$ there exists a piecewise smooth path $\...
4
votes
1answer
76 views

For which forms $F$ is the following volume finite?

Let $F \in \mathbb{R}[x_1, \cdots, x_n]$ be a homogeneous polynomial with degree $d \geq 2$. Put $$V(F) = m(\{(x_1, \cdots, x_n) \in \mathbb{R}^n : |F(x_1, \cdots, x_n)| \leq 1\})$$ where $m$ ...
1
vote
1answer
31 views

Hermitian forms with real coefficients

Let $z$ be the $n$-tuple of complex variables $(z_1,\ldots,z_n)$ and define $H:{\bf C}^n\to{\bf R}$ by $ H(z)=\sum_{i=1}^p |P_i(z)|^2 - \sum_{j=1}^q |Q_j(z)|^2, $ where $P_i,Q_j \in {\bf R} [z_1,\...
8
votes
2answers
222 views

Boundary triangulation induces triangulation

In $R^n$ (the real space) we have an open connected set $D$, such that $\partial D$ is triangulable. Can we prove the closure $\bar{D}$ is triangulable or any counterexample? Furthermore, the $\...
-1
votes
1answer
96 views

Orthogonal polynomials of the second kind

Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...
4
votes
0answers
72 views

Semi-algebraic approximation of maps

These are really two questions but I hope that the same method will solve both of them. For the purpose of this question let us fix a real closed field $R$, a bounded semialgebraic set $X$ over $R$, $...