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
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Homotopy equivalence of stably equivalent semialgebraic sets
In his book [1], Richter-Gebert introduces a notion of stable
equivalence for primary basic semialgebraic sets (subsets of
$\mathbb{R}^n$ defined by a conjunction of polynomial equations
and strict ...
1
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0
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Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure
In this question, the following fact was used by the respondent
A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar
measure contains a coset of $G^0$, the connected component of
$G$ ...
6
votes
1
answer
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Convex hull of a variety in real space
I am a physicist currently working on a question posed as part of an algebraic geometric description of a physical set:
I did not find a question that is closely related to what I am searching for yet,...
2
votes
1
answer
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If a variety over a real closed field has finitely many points they are singular
Let $F$ be a real closed field. Let $X$ be a positive-dimensional algebraic variety over $F$.
If $X$ finitely many $F$-points are they all singular?
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0
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Delta distributions that are smooth on strata of a singular manifold
This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
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forms on singular spaces that can be integrated on an LCI
I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real ...
2
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0
answers
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Characterizing non-zero polynomials on semialgebraic sets: a kind of positivstellensatz generalization
A polynomial positivstellensatz is an algebraic characterization of polynomials which are positive on a semialgebraic sets. Is there a similar kind of characterization which can determine whether a ...
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2
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Bialynicki-Birula decomposition for real analytic varieties
Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action ...
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2
answers
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Invariant theory over $\mathbb R$
$\DeclareMathOperator\SO{SO}$Suppose we have a (continuous) linear action of $\SO(n,\mathbb R)$ on a vector space $\mathbb R^N$. Consider the ring of invariants $A\subset \mathbb R[x_1,\ldots, x_N]$, ...
4
votes
1
answer
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Classes of curves closed under Minkowsky sum
Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological ...
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0
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Do singularities in real algebraic varieties have measure zero?
This question is related to Do proper Zariski closed sets of algebraic sets have measure zero
For algebraic varieties on complex numbers, it is easy to see that the locus of their singularies are set ...
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2
answers
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Nowhere negative polynomials form a semialgebraic set
Let $P_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$.
Is the subset $S\subset P_{d, n}$ of nowhere negative polynomials semialgebraic?
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votes
1
answer
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Real analyticity of continuous function via restriction to analytic curves
Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic)
Let $x\in X$ be a point and let $F: X\to \...
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votes
1
answer
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Is a complex or real algebraic variety homotopically equivalent to a CW complex?
Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers.
Let $X$ be an algebraic variety over $k$, say, quasi-projective and smooth (but not necessarily ...
7
votes
0
answers
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Geodesics on algebraic manifold
A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual embedded algebraic manifold) $M \subseteq \Bbb{R}^n$ that is also a nonsingular algebraic set (which means $...
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1
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An almost complex structure on the real $n$-sphere $S^n$
If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...
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0
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Degree of polynomials describing projection of algebraic set
Consider an algebraic subset $V\subseteq \mathbb{R}^{n+1}$ defined as the zero set of polynomials ${f_i}$ and the projection map $\pi: \mathbb{R}^{n+1}\to \mathbb{R}^n$ deleting the last entry.
By the ...
2
votes
0
answers
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Uniform Łojasiewicz constant in 2D
Łojasiewicz inequality is a classical result in real algebraic geometry. In particular, for any given polynomial $f:\mathbb R^2\to \mathbb R$ there is some $C>0$ and some $\alpha>0$ such that ...
2
votes
2
answers
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Rational bijections $\mathbb R\to(0;1)$ [closed]
Notation:
$$ (0;1)\ :=\ \{x\in\mathbb R:\ 0<x<1\}$$
There are simple rational stretches $\ f \colon (0; \, 1)\to\mathbb R,\ $ e.g. let $\ s\in(0;\, 1);\ $ then
$$ f(x)\ :=\ \frac{1-s}{1-x}-\...
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0
answers
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Lower bounds on lengths of sum-of-squares representations of particular polynomials
I am looking for literature on the problem of finding minimal (in the sense of number of terms) sum-of-squares representations of particular non-negative multivariate polynomials with rational ...
2
votes
0
answers
129
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Complexity of polynomial inequalities
What is known about the complexity of deciding whether a finite set of polynomial inequalities in $n$ real variables with integer coefficients is satisfiable? Decidability is guaranteed by Tarski's ...
1
vote
1
answer
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Product of subgroups of $SU(8)$ algebraic set?
Consider the special unitary group SU(8) acting on $\mathbb{C}^8\stackrel{\sim}{=}(\mathbb{C}^2)^{\otimes 3}$.
In particular, I am interested in the two subgroups $G_1=\mathrm{id}_{\mathbb{C}^2}\...
8
votes
2
answers
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Antiderivative of totally real polynomials
Let us say that a polynomial with real coefficients is totally real if all its complex roots are real and distinct. Let $P \in \Bbb R [X]$ be totally real. Is it true that
$$Q(X)=\int_0^XP(t)\,dt+aP(X)...
6
votes
1
answer
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Subsets of a ball/sphere with the largest sum of distances
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B_d$ and $S_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B_d$ is ...
1
vote
1
answer
252
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Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos?
I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form
$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$
where $\mu_{ijkl}$ are the ...
11
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0
answers
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Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
6
votes
1
answer
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Constructing M-curves à la Hilbert
I have been reading some text about Harnack's theorem. The theorem basically says that for degree $d$, the maximal number of connected components in the real (projective) plane of a plane curve with ...
6
votes
1
answer
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Decomposition of real algebraic varieties into manifolds
I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry.
I am dealing with a (real) algebraic variety $V$ of (Krull) dimension $n$. I keep reading ...
1
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0
answers
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Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?
I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
3
votes
0
answers
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The dimension of the $k$-independence
$\newcommand\Om\Omega$Let $(\Om,F,P)$ be a probability space. For some natural $n$, let $A_1,\dots,A_n$ be events, that is, members of the $\sigma$-algebra $F$. For $k\in[n]:=\{1,\dots,n\}$, these ...
22
votes
0
answers
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Knots realized as algebraic curves
Two questions:
Q1. Have researchers worked out minimum-degree
real algebraic curves in $\mathbb{R}^3$ realizing specific knots?
Some work on the trefoil is reported in this MSE question.
&...
5
votes
5
answers
615
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Elementary inhomogeneous inequality for three non-negative reals
I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
1
vote
0
answers
28
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Stratification which makes the defining functions isotrivial
Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic ...
7
votes
1
answer
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An elementary inequality for three complex numbers
The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
2
votes
0
answers
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About the Lipschitz property of stratified analytic, arc-analytic and semianalytic map
Let $f:U\to V$ be a $1:1$ map between open subsets $U,V\subset\mathbb{C}^N$ such that $f$ and its inverse $f^{-1}$ are both arc-analytic, semianalytic and piece-wisely real analytic with respect to ...
1
vote
1
answer
93
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Weak submodularity for consecutive indices
Let $f\colon \mathbf{R} \times \mathbf{R}^+ \rightarrow \mathbf{R}$ be defined by $f(x,y) = \frac{x^2}{y}$. Let $X = \left\lbrace x_1, \dots, x_n\right\rbrace \subseteq \mathbf{R}$, $Y = \left\lbrace ...
2
votes
0
answers
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Variation of Morse Functions: a reference request
Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...
2
votes
0
answers
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Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)
Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
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Generalization of: The dimension of a projective $\mathbb{F}$-variety equals the smallest codimension of a disjoint linear subspace
Let $\mathbb{F}$ be an algebraically closed field. Consider the following definition of the dimension of a (quasi)projective $\mathbb{F}$-variety, given in Harris Algebraic Geometry: A First Course:
...
1
vote
1
answer
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Identity map minus Cremona transformation
Let $ \delta $ be the triangle with vertices $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $\mathbb R^3$. It's a face of the standard octahedron. The Cremona transformation
$$\mathcal C: (x, y, z) \mapsto - \...
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votes
0
answers
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Distribution of rational points in the real locus of a planar algebraic curve
Let $C$ be a smooth projective geometrically connected curve over $\mathbb{Q}$. Assume that $g(C)=3$ and that $C$ is not hyperelliptic. Then the canonical sheaf defines a closed immersion $C\to\mathbb{...
1
vote
0
answers
331
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How to solve a system of quadratic equations?
Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$
$$ \left\{
\begin{array}{lr}
\mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, ...
2
votes
1
answer
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views
Finding the dimension of the intersection of two real algebraic varieties
Suppose we have two polynomials $p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional ...
5
votes
0
answers
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Hilbert scheme of real curves
Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise.
Let $H$ be the Hilbert ...
2
votes
0
answers
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A structure sheaf for real analytification of semialgebraic sets in the context of signed tropicalization
Let $X=Spec(A)$ be an affine scheme, where $A$ be a commutative algebra over a non-archimedean valued field $K$. Assume that $K$ is a real closed field with the unique ordering $<$, which should be ...
18
votes
1
answer
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"Real algebraic varieties" vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points
This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field....
17
votes
1
answer
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Any real algebraic variety is diffeomorphic to a real algebraic variety defined over $\mathbb{Q}$
Given a smooth proper real algebraic variety can you find a smooth proper real algebraic variety defined over $\mathbb{Q}$ that is diffeomorphic to it?
5
votes
2
answers
429
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The first part of the Hilbert sixteenth problem for elliptic polynomials
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.
Inspired by the first part of the Hilbert ...
11
votes
1
answer
477
views
A property of varieties between unirational and retract rational
EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open.
Let $V$ be a geometrically integral variety over a field $K$.
I consider the following ...
12
votes
6
answers
2k
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Differentiability of eigenvalues of positive-definite symmetric matrices
Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...