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 ...

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### 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 ...

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### 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, ...

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### 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 $...

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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 $...

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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}
...

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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 ...

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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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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 ...

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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?
...

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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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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 ...

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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 ...

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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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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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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) > ...

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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 \...

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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{...

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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) \, \, \, \, , \, \, ...

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### 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).
...

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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 ...

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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 $f$ ...

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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 ...

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### 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(\...

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### 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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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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### 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 $\mathbf{v}$...

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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 ...

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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 ...

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**1**answer

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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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 ...

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**1**answer

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 $...

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**1**answer

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 ...

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**1**answer

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
$$\...

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### 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 ...

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### 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) ...

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**1**answer

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 ...

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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 ...

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### 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 ...

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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 ...

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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. ...

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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 ...

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**1**answer

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 $\...

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**1**answer

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 $m$ ...

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**1**answer

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,\...

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**2**answers

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 $\...

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**1**answer

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 ...

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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 $R$, $...