# 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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### 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-...
1 vote
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1 vote
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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 ...
86 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 ...
65 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 ...
1 vote
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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 ...
50 views

### 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 ...
33 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 ...
94 views

### 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 ...
54 views

### Semi-algebraic set has nonempty interior relative to Zariski closure

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 ...
1 vote
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### Structure of Closed Semialgebraic set

I am trying to prove the following, from Benedetti and Risler's book: The "above proposition" is: It seems an easy proposition that boils down to taking the complement of the complement of ...
1 vote
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### Upper bound for amount of intervals in intersection of interval sets

I have two sets of numbers which are unions of disjoint intervals, and I have to find an upper bound for how many of such intervals can there be in the intersection of the two sets. Here's a diagram ...
154 views

### Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if ...
145 views

### The quotient scheme $X/\Gamma$ when $X$ is separated and every orbit is contained in an affine.

I am trying to solve Problem II.4.7(a) of Hartshorne: The only candidate I can think of for $X_0$ would be the quotient scheme $X/\sigma$. If it exists, it must be unique by the usual argument. First ...
1 vote
### When does the (real) zero locus $F(x,y,t) = 0$ "look the same" for all values of the parameter $t$?
Disclaimer: I am not a mathematician by training. I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous ...