# 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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### Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
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Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
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### Sufficient condition for pair of real quadrics to have real intersection

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero. Let $q$ be a real quadric. Then $q$ has a real zero if and only if $q$ has indefinite ...
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### Real-isability of a (relatively small) subconfiguration of the Klein configuration

The Klein configuration consists of $60$ points and $60$ planes in $\mathbb C\mathbf P^3$, each point lying on $15$ of the planes and each plane containing $15$ of the points. It appears, among many ...
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### How small need a perturbation be to not change the diffeomorphism type of a variety?

Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$. Assume that $X$ is smooth and has codimension $1$. Then ...
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### An inequality problem for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...
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### An inequality for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...
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### Consequences of Nash-Tognoli Theorem

The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact. However, I am not aware of ...
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### Is a continuous rational function Lipschitz?

Let $f\in \mathbb{R}(x_1,\ldots,x_n)$ be a rational function. Suppose that $f$ is continuous on $\mathbb{R} ^n$. Must it be Lipschitz on the unit ball? This question might be related to Are continuous ...
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### Degree four polynomials with no real roots

Consider a degree four polynomial $$f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x]$$ with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...
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### Monge–Ampère equation with polynomials

Let $P$ be a real polynomial of $n$ variables. Does the Monge–Ampère equation $$\det D^2 Q=P$$ always have a (global) real polynomial solution $Q$? I think this should be something standard, but I ...
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### A category of spaces in which $\mathbb{R}_{>0}$ is not isomorphic to $\mathbb{R}$

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$I am writing up notes on totally positivity in flag varieties. I often have the following situation: I have a complex variety $X$ defined over $\RR$ and a real ...
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### Is the Segre embedding of two real varieties a real variety?

$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve ...
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### Topology types in families of real or complex varieties

In René Thom, "Structural Stability and Morphogenesis" on p. 21ff there is the following statement: Let $$P_j(x_i,s_k) = 0$$ be a set of polynomial equations over the real or complex numbers,...
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### Multivariate polynomials with given real zeros

Given natural numbers $N$ and $n$, I am looking for a family $P_{Nn}$ of polynomials in $n$ variables with the following properties: (i) Every polynomial in $P_{Nn}$ is determined by the function ...
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$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$. Let $d$ denote the Euclidean distance in $\R^n$. Do then ...
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### Computer algebra tools for finding real dimension of an algebraic variety

I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set? The CAD-based ...
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### Plot two implicit surfaces in 3D and highlight their intersection [closed]

I want to plot the two surfaces which are defined in $\mathbb{ R }^3 \ni ( x, y, z )$ via the equations $0 = y^2 - x*(x^2 + 1)$ and $0 = z^2 - y*(y^2 + 1)$, respectively. Moreover, I want also ...
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### An inequality in four variables

Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that ...
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### Finite number of topological spaces realized by varieties of bounded degree?

I am not familiar with algebraic geometry so I am sorry if this question is terribly ignorant. Any basic reference is appreciated. Is there a finite bound on the number of topological spaces that can ...
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### On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
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### Certificates of connectivity of basic semi-algebraic sets

Given real polynomials $p_1, \ldots, p_n \in {\mathbb R}[x_1, \ldots, x_d]$, consider the closed basic semi-algebraic set $S \subseteq {\mathbb R}^d$ given by S := \{x \in {\mathbb R}^d : p_i(x) \...
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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 ...
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1 vote
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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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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?