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).
312
questions
7
votes
1
answer
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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 ...
2
votes
1
answer
105
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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 ...
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:
$$\sum_{cyc}\...
16
votes
1
answer
562
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Approximating zero sets of real polynomials with "less complicated" polynomials
Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
6
votes
1
answer
309
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On strict positivity and Schmüdgen's Positivstellensatz
Schmüdgen's Positivstellensatz requires the polynomial to be strictly positive on a semialgebraic set. While trying to understand it, I am just wondering if the strictly positive condition can be ...
23
votes
1
answer
744
views
Is every minimal hypersurface in $S^n$ algebraic?
Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question
Is every minimal ...
1
vote
1
answer
305
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A close-form solution for a simple quadratic optimization problem
Is there any closed form solution for the following optimization problem:
\begin{align}
&\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\...
9
votes
1
answer
510
views
Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
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 ...
2
votes
2
answers
343
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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 $\...
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 $\...
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 ...
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 $A(2,0)$?
2
votes
1
answer
255
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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 ...
2
votes
1
answer
380
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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 ...
3
votes
2
answers
425
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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,\...
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 ...
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 ...
3
votes
1
answer
126
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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 ...
2
votes
1
answer
166
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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 ...
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 ...
2
votes
1
answer
240
views
Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
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 ...
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 ...
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 ...
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 ...
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 $\...
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 \...
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 $...
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 ...
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$...
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 $\...
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 "...
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 ...
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 ...
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 ...
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, ...
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 $...
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 $...
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}
...
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 ...
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 ...
7
votes
1
answer
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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,...
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?
...
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}{\...
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 ...
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 ...
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 $...
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(...
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) > ...