# 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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### 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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### 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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### 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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### 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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### 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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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 $$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ... 0answers 19 views ### Stratification which makes the defining functions isotrivial Let 0\in X\subset\mathbb{C}^N be a germ of complex space and 0\in Z\subset X be a closed analytic subset (globally) defined by holomorphic functions f_1,\dots,f_r. Is there a complex analytic ... 1answer 287 views ### An elementary inequality for three complex numbers The following problem arose in asymptotic analysis of difference equations. Numerical maximization suggests that for all nonzero complex numbers a,b,c we have$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
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 ...