# Questions tagged [real-algebraic-geometry]

Real algebraic geometry is the study of algebraic geometry over the real numbers, or more generally formally real (esp. real closed) fields. Problems in this tag may require a mix of methods from algebraic geometry and techniques from o-minimal (esp. semialgebraic) geometry.

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### System of $n$ simultaneously diagonal real quadrics in $n+1$ variables has all solutions real

A single isotropic real quadric in 2 variables always has 2 (projective) real solutions. See Zero set of system of two real quadratic forms for the explicit form. I've noticed that a system of 2 ...
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### When are the zeroes from Bezout's theorem real/rational/integral

Suppose I have $n$ quadratic homogeneous polynomials $f_1, \cdots, f_n$ in $n+1$ variables over the field $\mathbb{C}$. Bezout's theorem says that generically there will be $2^n$ common zeroes. I was ...
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### Help me complete a proof about the enumeration of similar intersections of an algebraic surface with a hyperplane

This is something of a follow-up to this question that i posted almost a year ago: What are the conditions on two multivariate real polynomials $f$ and $g$ so that their zero sets are similar? It was ...
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### Question on matrix spaces that is derived from several vector spaces

Suppose I have $m$ column vectors $x_i \in X \subset \mathbb{R}^n$, $i \in \{1,\ldots,n\}$. If I construct a matrix $\mathbf{x}= [x_1^T ; \ldots ; x_m^T]$, then what set does $\mathbf{x}$ live in ...
1 vote
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### What is the minimal number of polynomial squares needed to represent any sum of any number of polynomial squares?

Let $Q^{(n, d)}$ be a set of real polynomials in $n$ variables of degree $d$, and $S_k^{(n, d)}$ the set of the sum of squares of any $k$ such polynomials: \begin{equation} S_k^{(n, d)} := \left\...
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### Proof of uniqueness of real-closure of an ordered field

I'm reading the proof of uniqueness of real-closure of an ordered field $F$, that is an algebraic extension $R$ of $F$ such that $R$ is real-closed and the unique order on $R$ extends that of $F$. I ...
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### Theorems of calculus for real-closed fields [duplicate]

Theorems such as Intermediate value theorem Rolle's theorem Mean value theorem Positive/negative derivative implies strictly increasing/decreasing Hold for polynomials over real-closed fields. If ...
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### Defintion of a real algebraic space in Atiyah's K-theory and reality

In Atiyah's paper "K-theory and reality", p. 370, there is an example of a "real" algebraic space. Given the complex projective space $X=P(C^n)$, one considers the standard line-...
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### Are these definitions of dimension for a real algebraic variety all equivalent?

I would like to believe that the following notions of dimension are all equivalent for an a real algebraic variety in $\mathbb{R}^n$. Lower Minkowski dimension Upper Minkowski dimension Hausdorff ...
196 views

### Dimension of a semialgebraic set equals the dimension of its closure.

I am trying to prove that if $X \subset \mathbb{R}^{n}$ is semialgebraic, then $\operatorname{dim}X = \operatorname{dim}\overline{X}$, where the dimension of a semialgebraic set is the supremum of the ...
120 views

### Product of semialgebraic sets is semialgebraic

I am trying to prove that the product of semialgebraic sets is semialgebraic. If $X \in \mathbb{R}^{n}$ and $Y \subset \mathbb{R}^{m}$ are semialgebraic, I can't see the polynomial conditions involved ...
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### Real radical ideal which is not prime

Terminology: Let $I\subset\mathbb{R}[X_1,...,X_n]=:A$ be an ideal. We call $\sqrt[\mathbb{R}]{I}:=\{f\in A\ |\ \exists k\in\mathbb{N},\ g_1,...,g_m\in A:\ f^{2k}+\sum_{i=1}^mg_i^2\in I\}$ the real ...
1 vote
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### Semialgebraic set technical lemma

Suppose that $U\subseteq \mathbb{R}^3$ is real-semialgebraic set such that $U'\times \mathbb{R}\subseteq \overline{U}$ where $U'$ is an infinite set. I want to show that there exist $x,y$ such that ...
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### Parametrizing an algebraic curve

Suppose $C\subset \mathbb{R}^3$ is a path-component of an irreducible real-algebraic curve. Is there a smooth parametrization that covers all but possibly one or two points of $C$? If not, is there a ...
58 views

### Number of intersection points of plane and algebraic curve

Suppose $C\subseteq \mathbb{R}^3$ is an irreducible real-algebraic curve of degree $k$ and $P\subset \mathbb{R}^3$ is a plane. Suppose that $C$ intersects $P$ finitely many times. What is the best way ...
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### Algebraic curve contained in plane or only intersects it finitely many times

Suppose that $C\subset \mathbb{R}^3$ is an irreducible real-algebraic curve that meets a plane infinitely many times. Is it true that $C$ must be contained in this plane? I'm working on a paper that's ...
Suppose $X\subseteq \mathbb{R}^3$ is a real semi-algebraic set. Consider the subspace topology induced by the Euclidean topology on the Zariski closure $\overline{X}.$ Can we guarantee that $X$ has ...