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Questions tagged [real-algebraic-geometry]

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0
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1answer
60 views

On solution to the equation $x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}\left(\sum_{i=1}^{6}a_{i}x_{i}\right)^{2}=1$

For any $a_{1}, a_{2}, \dots, a_{6} \in \mathbb{R}$ with $$\sum_{i=1}^{6}a_{i}^{2}=1$$ is it true that there always exist $x_{1}, x_{2}, \dots, x_{6} \in \mathbb{R}$ with $\displaystyle\sum_{i=1}^{...
0
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0answers
16 views

Quadratic forms which represent the same element

Let $a,b \in F^x$. Show that for quadratic forms holds: $D(\lt 1,a \gt) \cap D(\lt 1,b \gt) \subseteq D(\lt 1,-ab \gt)$ Here these sets represent the set of elements in $F^x$ which are represented ...
1
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1answer
13 views

Existence proof for the minimal $n \in \mathbb{Z}$ that satisfies $mb \leq n+1$ for a general field $K$

I am studying real algebraic geometry. Let $K$ be an Archimedean Ordered field. While trying to prove this lemma, I am not able to understand how can we choose a minimal $ n \in \mathbb{Z}$ which ...
1
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1answer
17 views

Show that quadratic form is anisotropic over rational numbers

How to determine if quadratic form $q(x,y,z)=-6x^2+ \frac{7}{2}y^2-\frac{25}{7}z^2$ is anisotropic over $\mathbb{Q}$? This quadratic form is a diagonalisation of another quadratic form. Is there any ...
2
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1answer
62 views

If $\omega \in \bigwedge^k\mathbb{R}^d$ is decomposable over $\mathbb C$, is it decomposable over $\mathbb R$?

Let $k,d$ be positive integers, and let $\omega \in \bigwedge^k\mathbb{R}^d$ be decomposable in $ \bigwedge^k\mathbb{C}^d$. Is $\omega$ decomposable in $\bigwedge^k\mathbb{R}^d$? Edit: Let me be ...
2
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0answers
48 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This question was inspired by this beautiful answer: Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ ($1 \le i_1 < \ldots < i_2 \le 4$) be a basis for $\bigwedge^2V$, ...
4
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1answer
66 views

Characterising minors of diagonal matrices

Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$. Are ...
7
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0answers
73 views

When do polynomial equations come from complexification?

If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,...
1
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1answer
62 views

When do two plane cubic curves have 9 real intersection?

What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an ...
2
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0answers
83 views

Concerning the ring of continuous functions on $\mathbb{R}$

It is not difficult to check that the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a ring (an $\mathbb{R}$-algebra), and similarly (if I am not wrong), the set of continuous ...
2
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1answer
80 views

Denseness of algebraic points within a variety

I want to use the following statement concerning varieties, but I do not know why it is true. Claim. Let $V \subset \mathbb{C}^n$ be a variety defined over $\mathbb{Q}$. Then the set $V \cap \bar{\...
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0answers
47 views

Belonging to the same connected component of a semialgebraic set

Warm-up and main questions: Let $X \subset \mathbb{R}^n$ be a semialgebraic set and $x_1, x_2 \in X$. How can I effectively check if $x_1$ and $x_2$ are in the same connected component of $X$? Let $f ...
4
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2answers
127 views

Simultaneous real solution of $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$

I am trying to solve the following system of non-linear equations in real numbers: $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$, with $x,y$ real. I can only see that $xy\ne 0$. I have no clue whether a ...
4
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1answer
109 views

Is the minimizer of the distance from a point to a closed set generically unique?

Let $\mathcal{C} \subset \mathbb{R}^n$ be a closed set and let $E$ be the set of points in $\mathbb{R}^n$ for which there is not a unique closest element of $\mathcal{C}$. That is, if $x \in E$, then ...
2
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1answer
67 views

Algebraic geometry over general fields via Galois theory

I was informed by my supervisor that it is possible to study algebraic sets over non-algebraically closed fields by passing to an algebraically closed setting via Galois theory. How exactly do we do ...
3
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0answers
104 views

General techniques for proving coefficients of a multinomial are all positive

I have encountered a problem in my research where I need to prove that all the coefficients of a certain multinomial of the form $R_k(\mathbf{x}, \mathbf{y}) = \frac{P_k(\mathbf{x}, \mathbf{y})}{Q_k(\...
1
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0answers
28 views

Computing locus of points with positive dimensional fibers

I became interested in the following problems by studying kinematic configuration spaces: Let $C=V(f_1,\ldots,f_r)$ where $f_i\in \mathbb{R}[\vec{x},\vec{y}]$, $J=V(g_1,\ldots,g_s)$ where ...
1
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0answers
68 views

System of $n+1$ polynomial equations in $2m$ unknowns

I aim to study the following system of equations with respect to $(x_{1}, \dots, x_{m}, y_{1}, \dots, y_{m})$ $$ \left\{ \begin{array}{c} y_1x_1+y_2x_2+y_3x_3 + \dots + y_mx_m=c_1 \\ y_1x_1^2+...
1
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0answers
27 views

An affine $G$ such that $G(p)_x(\mathbb{R}^2) \geq 0$, $p \in \mathbb{R}[x,y]$ is homogeneous of odd degree

Let $p\in \mathbb{R}[x,y]$ be a homogeneous polynomial of odd degree $d \geq 1$. Can one find an affine $\mathbb{R}$-algebra automorphism $G$ of $\mathbb{R}[x,y]$ such that $G(p)_x(\mathbb{R}^2) ...
0
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0answers
136 views

Homogeneous polynomial of even degree in two variables

Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$. Page 281 says: "Let us denote $\nabla(n, d, \mathbb{C}) \subset ...
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0answers
34 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 ...
0
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0answers
42 views

Existence of a real solution to a certain system of inequalities— generalized

The current question is a generalization of this question. Let $x,y,z,w$ be four variables over $\mathbb{R}$. Let $A,B,D,E \in \mathbb{R}[x,y,z,w]$ be four homogeneous polynomials, each of degree $5$....
10
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3answers
406 views

Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$ $$ P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij}, $$ I ...
1
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0answers
37 views

Algebraic curves of odd degree - only finitely many lines through the origin fail to intersect

Let $f$ be a curve of odd degree $d$ in $\mathbb{R}^2$ with $f(0,0) \neq 0$. Show that there are at most finitely many real numbers $\lambda$ for which $f$ fails to intersect the line $y=\lambda x$. ...
0
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0answers
25 views

(Euclidean) open orbit in an irreducible algebraic set of $\mathbb{R}^{n}$

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 algebraic ...
0
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1answer
34 views

Is the image of the map $A \to \bigwedge^{k}A $ from matrices of a given rank closed?

$\newcommand{\Cof}{\operatorname{cof}} \newcommand{\id}{\operatorname{Id}}$ Let $V$ be a real $d$-dimensional vector space ($d>2$). Let $2 \le k \le d-1$ be fixed, and let $r>k$. Define $H_r=\{ ...
2
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1answer
56 views

Is union of matrices of different ranks a submanifold?

Let $V$ be a real $d$-dimensional vector space ($d>2$). Let $1 \le r_1 <r_2 < \dots < r_k = d$ be a fixed finite sequence of numbers. Define $H=\cup_{i=1}^kH_{r_i}$, where $H_{r}=\{ A \in ...
0
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0answers
54 views

Disjoint Union Basic-Closed Semialgebraic Sets

A basic-closed semialgebraic set in $\mathbb{R}^{n}$ is defined as $$ M=\lbrace x \in \mathbb{R}^{n} \mid f_{1}(x) \geq 0, \ldots, f_{m}(x) \geq 0 \rbrace$$ for some polynomials $f_{i}$ in $n$ ...
1
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1answer
87 views

Lebesgue measure of a semi-algebraic set

Let $A$ be a semi-algebraic subset of $\mathbb{R}^n$ that is semi-algebraically homeomorphic to $(0,1)^k$ with $k<n$. I would require a result stating that $\mathcal{L}^n(A)=0$ where $\mathcal{L}^...
2
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0answers
78 views

Why is the set of all $x \in \mathbb{R}$ satisfying $ x^2 - 8x + 15 \leq 0 $ representable as a polyhedron? [closed]

I was attempting this tutorial question: a) For each of the following sets, decide whether it is representable as a polyhedron. (ii) The set of all $x \in \mathbb{R}$ satisfying $$x^2 − 8x + ...
1
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1answer
61 views

Characterizations of bounded real zero sets

Given a multivariate polynomial $p(x_1,\ldots,x_n)$ with real coefficients, its real zero set, i.e. the set of real roots of the polynomial, is $\{(x_1,\ldots,x_n) \in \mathbb{R}^n \mid p(x_1,\ldots,...
3
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0answers
56 views

How to proof a real variety is not locally a manifold

I'm becoming desperate with a problem, which doesn't seem so difficult. I have a real algebraic set $X$ in $\mathbb{R}^{15}$ defined by $$ x_1^2 + y_1^2 + z_1^2 - 1 = 0\\ x_2^2 + y_2^2 + z_2^2 - 1 = ...
2
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0answers
81 views

Does an irreducible real affine algebraic set/ its complement has finitely many connected components in the Euclidean topology?

Let $n\ge 2$ and $V$ be an irreducible affine algebraic set in $\mathbb R^n$ . Then is it true that $V$ has only finitely many connected components in the Euclidean topology of $\mathbb R^n$ ? Does $\...
1
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2answers
277 views

How can one prove that this polynomial is non-negative?

How one can prove the following inequality? $$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ I plotted the graph on Wolfram Alpha and found that the inequality seems to ...
0
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1answer
31 views

Is a bilinear form with positive $k$-minors positive-definite?

This question is related to this one. Let $V$ be a $d$-dimensional real vector space, and let $g:V \times V \to \mathbb{R}$ be a symmetric bilinear form. Suppose that for every sequence of vectors $...
1
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1answer
62 views

What geometry (if any) studies sets defined by infinitely many polynomial inequalities?

Question: Semi-algebraic geometry studies the solution sets of finitely many polynomial inequalities (in $\mathbb{R}^n$). What field (if any) could be considered the study of the solution sets of ...
2
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1answer
48 views

Representing a co-Heyting algebra as a lattice of semialgebraic sets in a real algebraic variety

Every co-Heyting algebra can be embedded in the co-Heyting algebra of closed sets in a spectral space. (Co-heyting algebras are the order-theoretic duals of Heyting algebras.) Now I am told the ...
3
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2answers
87 views

Boundary of the cone of positive polynomials

Assume that $P^d(\mathbb{R})$ is the cone of polynomials $f\in\mathbb{R}[x_1,...,x_n]$ such that $\deg f\leq d$ and $f(x)\geq 0$ for all $x\in\mathbb{R}^n$. Is it true that the boundary of this cone ...
0
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1answer
47 views

Points and dual $\mathbb{R}$-algebra to $\mathbb{R}[V]$ on real affine variety $V$

Let $V$ be a real affine variety with its ring $\mathbb{R}[V]$ of regular functions. Denote by $|\mathbb{R}[V]|$ the set of all $\mathbb{R}$-algebra homomorphisms $h:\mathbb{R}[V]\to \mathbb{R}.$ Any ...
2
votes
1answer
67 views

Geodesic through/between singularities

Let $X\subseteq\mathbb R^q$ be a (singular) real algebraic set, and let $$ g\colon[0,T]\to X $$ be a geodesic (that is, a shortest path between its ends). Is it true that the image $g([0,T])$ ...
1
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0answers
50 views

Convergence of sum of squares relaxations for global polynomial optimization

I have studied and understood the Moment-SOS hierarchy proposed by Lasserre where a sequence of semidefinite programs are required to be solved and a rank condition for the moment matrices is invoked ...
4
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0answers
191 views

Sum-of-squares quartic polynomials in three variables

Suppose $f \in \mathbb R [x,y,z]$ is a homogeneous polynomial of degree $4$. Furthermore, suppose we can write $$f(x,y,z)=\sum_i p_i(x,y,z)^2$$ where each $p_i$ is a homogeneous polynomial of degree ...
3
votes
1answer
79 views

Existence of Countably Infinite Varieties Over Uncountable Fields

Let $k$ be an uncountable field. Can a countably infinite set in $k^n$ be obtained as the zero set of a system of polynomial equations? What if $k$ is algebraically closed or a real-closed field? I ...
2
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0answers
53 views

A confusion on the fact that the set of real rational function are not archimedean

In the book of Real algebraic geometry by Bochnak-Coste-Roy, in page $7$ (and yes, I have stuck on the first page of the book :) ), it is given that There is exactly one ordering $\mathbb{R} (X)$ ...
0
votes
1answer
38 views

A confusion about the order in the set of rational functions

In the book of Real Algebraic Geometry by Bocknak-Coste-Roy, at page $7$, it is given that There is exactly one ordering $\mathbb{R} (X)$ such that $X$ is positive and smaller than any positive ...
2
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0answers
114 views

One question about semi-algebraic variety

We identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, and let $C\subset \mathbb{C}^n$ be a semi-algebraic curve, then can we always find a complex algebraic curve $V'\subset \mathbb{C}^n$ containing $C$? ...
0
votes
1answer
35 views

Bivariate Form Sum of Squares

Let $F \in \mathbb{R}[x,y]$ be nonnegative and homogeneous of degree $2n$. Then it can be written as a sum of two squares.
0
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1answer
105 views

Zero set of homogeneous irreducible polynomial over reals

To show that the zero set of an irreducible polynomial in $k[X_1,\dots,X_n]$ is irreducible in $\mathbb{A}^n_k$, we need to assume that the field $k$ is algebraically closed. Indeed, for $f(x,y)=(x^2-...
1
vote
1answer
63 views

Given a polynomail complete intersection in $\mathbb{RP}^n$ how can I figure out the topology?

I want to play around with real-algebraic geometry, but am not sure where to start. I have a couple basic questions which motivate my study: How can I come up with explicit models for smooth surfaces ...
2
votes
1answer
64 views

Is the gaussian function a Nash function on $\mathbb{R}$?

Let me explain the question. Take a gaussian function $f:t\in \mathbb{R} \mapsto \exp(-t^2)$. Does there exist a nontrivial polynomial $P\in \mathbb{R}[X,Y]$ such that $$ \forall t\in \mathbb{R},\...