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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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 $G$ ...
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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,...
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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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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 ...
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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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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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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 $A\subset \mathbb R[x_1,\ldots, x_N]$, ...
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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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148 views

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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290 views

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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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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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 ...
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Algebraic properties of geodesics

This is a question related to my last post. I will use the same definition here. A complete smooth manifold $M$ with an affine connection $\nabla$ is said to have an algebraic g-model of dimension $n$ ...
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Geodesics on algebraic manifold

A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual algebraic manifold which is required to be embedded) $M \subseteq \Bbb{R}^n$ (with induced topology) that is ...
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1answer
877 views

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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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 ...
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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 ...
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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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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 ...
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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 ...
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1answer
144 views

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}\...
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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)...
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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 ...
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1answer
137 views

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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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.   &...
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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 ...
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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 ...
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287 views

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)...
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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 ...
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1answer
74 views

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 ...
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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 ...
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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: ...
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1answer
103 views

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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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{...
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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, ...
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1answer
179 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 ...
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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 ...
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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 ...
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1answer
555 views

"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....
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1answer
861 views

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?
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376 views

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 ...
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274 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 ...
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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 ...

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