Questions tagged [real-algebraic-geometry]

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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General Tarski-Seidenberg Theorem

The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of ...
KhashF's user avatar
  • 2,598
2 votes
1 answer
105 views

When is a set defined by multivariate polynomial inequalities convex?

Consider the set of real numbers given by $$S = \{(a,b,c,d,e,f,g,h) \in [0,1]^8 : 0 \le \frac{e(g-h)}{b(g-f)} \le 1 \text{ and } 0 \le \frac{e(h-f)}{(1-b)(g-f)} \le 1\}$$ Note that this set can also ...
insert-name-here's user avatar
11 votes
3 answers
628 views

Polynomial inequality of sixth degree

There is the following problem. Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$ Prove that: $$\sum_{cyc}\...
Michael Rozenberg's user avatar
16 votes
1 answer
562 views

Approximating zero sets of real polynomials with "less complicated" polynomials

Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
SMS's user avatar
  • 1,293
6 votes
1 answer
309 views

On strict positivity and Schmüdgen's Positivstellensatz

Schmüdgen's Positivstellensatz requires the polynomial to be strictly positive on a semialgebraic set. While trying to understand it, I am just wondering if the strictly positive condition can be ...
SophiaA's user avatar
  • 61
23 votes
1 answer
744 views

Is every minimal hypersurface in $S^n$ algebraic?

Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question Is every minimal ...
JSCB's user avatar
  • 1,610
1 vote
1 answer
305 views

A close-form solution for a simple quadratic optimization problem

Is there any closed form solution for the following optimization problem: \begin{align} &\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\...
Math_Y's user avatar
  • 311
9 votes
1 answer
510 views

Do these surfaces intersect?

For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$ with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$, does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
mathers1's user avatar
3 votes
0 answers
279 views

Real analytic function: zero set of the gradient is a subset of the zero set of the function

I had this question when reading Bierstone and Milman's famous paper "Semianalytic and subanalytic sets". In their proof of the Łojasiewicz gradient inequality (Proposition 6.8 in the paper), they ...
user151260's user avatar
2 votes
2 answers
343 views

Why does a complex linear normalization of a real algebraic surface inherit a real structure?

Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.) (1) Let a surface $X$ in $\...
Mikhail Skopenkov's user avatar
7 votes
1 answer
537 views

What is the topology on the set of field orders

Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)? For example for the function field $\...
HenrikRüping's user avatar
3 votes
1 answer
324 views

Lower bound for polyhedral real quantifier elimination

All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$. Is there an example of double exponentiality with ...
VS.'s user avatar
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2 votes
2 answers
341 views

How to determine the range of values ​of A(i,j) in Covariance matrix A?

Let $A(i,j), i,j=0,1,2$ be the covariance matrix of three random variables. If we know all the entries except $A(2,0)$ and $A(0,2)$, how to determine the range of possible values of $A(2,0)$?
phybrain's user avatar
  • 103
2 votes
1 answer
255 views

Homogeneous polynomial in 4 variable with non degenerate zero

I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly ...
Gil Sanders's user avatar
2 votes
1 answer
380 views

Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid

This is a follow-up question, to a question I asked earlier. See Algebraic curve intersecting square-grid. Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space tightly packed in the ...
Till's user avatar
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3 votes
2 answers
425 views

Algebraic curve intersecting square-grid

Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells. Formally $$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$ for $i,j = 0,\...
Till's user avatar
  • 469
5 votes
0 answers
162 views

Example of applying real quantifier elimination algorithm for polynomials

Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
Evan's user avatar
  • 51
8 votes
0 answers
237 views

Monadic second-order theories of the reals

I’m looking for a survey of monadic second-order theories of the reals. I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
user avatar
3 votes
1 answer
126 views

Intersection of quadratic equations with planted solutions?

Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection? In general what can we say about intersection of $k$ quadratics? How many ...
Turbo's user avatar
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2 votes
1 answer
166 views

SDP representation of ideal polynomials for positivstellensatz refutations

If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e. $$ S = \{ x\in \mathbb{R}^n \ | \ h_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$ we can produce a ...
Andrea Olivo's user avatar
5 votes
1 answer
611 views

Polynomials (or analytic functions) vanishing on a real algebraic set

I have seen the following result stated several times in the literature, without proof: Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, and assume $P\in\mathbb{K}[X_{1},\ldots,X_{n}]$ is an ...
user111's user avatar
  • 3,761
2 votes
1 answer
240 views

Does quantifier elimination help here?

Suppose we have a quantified linear program $$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$ $$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$ $$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$ $$...
VS.'s user avatar
  • 1,816
0 votes
0 answers
59 views

A retract algebraic subset of the plane which does not admit an algebraic retraction

What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$? What ...
Ali Taghavi's user avatar
10 votes
1 answer
338 views

Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary. Baeza [B, V.6.3] extended this result to Witt ...
Uriya First's user avatar
  • 2,846
9 votes
0 answers
207 views

Kronecker's theorem in higher dimension

Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
François Brunault's user avatar
5 votes
0 answers
349 views

Are nearby points in an algebraic curve necessarily connected?

I would like a result of the following form: For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...
Dustin G. Mixon's user avatar
7 votes
1 answer
289 views

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

This is a cross-post. Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
Asaf Shachar's user avatar
  • 6,611
1 vote
0 answers
239 views

Orbits of unipotent groups

Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \...
Jakub Konieczny's user avatar
1 vote
0 answers
55 views

Sum of squared nearest-neighbor distances between points on the sides of a rectangle

For positive real numbers $a,b$, let $R$ denote the $a\times b$ rectangle $[0,a]\times[0,b]$. Let $A_1,\dots,A_4$ be points on the sides of $R$, one point on each side. For each $j=1,\dots,4$, let $...
Iosif Pinelis's user avatar
3 votes
0 answers
61 views

Biggest Cartesian Product Included in a Real Plane Curve

Suppose an irreducible smooth $p \in \mathbb{R}[x_1,x_2]$ is given, and we would like to find finite sets $S_1 , S_2 \subset \mathbb{R}$ such that $p(S_1 \times S_2)=0$ and $|S_1 \times S_2|$ is as ...
alpx's user avatar
  • 351
6 votes
1 answer
550 views

Number of connected components of degree 2 affine algebraic varieties

Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients $$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$ for $x\in\mathbb R^m$...
Dominik's user avatar
  • 83
2 votes
1 answer
159 views

About independence spread

$A$, $B_{i}$ are some events. If $A$, $B_{i}$ are independent $\forall i \in \mathbb N$ and $A \cap B_{1}, A \cap B_{2}, ..., A \cap B_{k}, ...$ are independent in aggregate, how to show, that $\...
Lisa's user avatar
  • 113
17 votes
1 answer
763 views

Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "...
Misha Verbitsky's user avatar
9 votes
0 answers
546 views

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 ...
Timothy Chow's user avatar
  • 78.1k
1 vote
0 answers
57 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 ...
Jakub Konieczny's user avatar
10 votes
2 answers
202 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 ...
Avi Steiner's user avatar
  • 3,031
7 votes
1 answer
131 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, ...
Vadim Ogranovich's user avatar
2 votes
0 answers
132 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 $...
Vadim Ogranovich's user avatar
11 votes
0 answers
225 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 $...
loup blanc's user avatar
  • 3,574
19 votes
2 answers
748 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} ...
David Zhang's user avatar
  • 1,292
19 votes
1 answer
691 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 ...
Michael Griffin's user avatar
5 votes
1 answer
108 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 ...
AleAlvAlwaysDIEZ's user avatar
7 votes
1 answer
2k views

Solving system of bilinear equations

Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system $$\left\{ x^T A_i y = b_i : i = 1,\dots,m \right\}$$ in variables $x,...
grok's user avatar
  • 2,489
3 votes
1 answer
171 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? ...
Katjoek's user avatar
  • 133
2 votes
1 answer
318 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}{\...
Iosif Pinelis's user avatar
8 votes
1 answer
437 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 ...
Iosif Pinelis's user avatar
4 votes
1 answer
313 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 ...
user237522's user avatar
  • 2,783
17 votes
1 answer
634 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 $...
Asaf Shachar's user avatar
  • 6,611
3 votes
0 answers
302 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(...
Asaf Shachar's user avatar
  • 6,611
4 votes
1 answer
211 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) > ...
Asaf Shachar's user avatar
  • 6,611

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