# Questions tagged [semialgebraic-geometry]

Semialgebraic geometry is the study of semialgebraic sets. This tag is intended for problems in (or relating to) semialgebraic geometry and its generalizations: semianalytic, subanalytic, and o-minimal geometries. These areas have strong connections to logic via the Tarski-Seidenberg theorem, and solutions to problems in this tag often involve a mix of geometric and logical arguments.

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### A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty

I would like to prove the following result : A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty The dimension here has to be understood in the semi ...
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### Proving that a set is semi algebraic using Hilbert’s basis theorem

I fix $\epsilon>0$ and consider the set $$A=\{(a,b+d,-b)\in\mathbb{R}^3 : a\in\mathbb{R}, b\in\mathbb{R}, d\in(0,\epsilon) \}$$ I want to prove the set is semi-algebraic. My idea is first to ...
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### Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
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### Semi algebraic sets and smooth manifolds

I would like to prove a result that I cannot see why it could be wrong in order to fix the idea. Here is the result : Let $A\subset\mathbb{R}^{n}$ be a semi algebraic set which is a smooth ...
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### Link between semi-algebraic dimension and vector space dimension

I am new to semi-algebraic geometry and I have encountered a very pleasant proposition to define the semi-algebraic dimension of a semi-algebraic subset $A$ of $\mathbb{R}^n$. This definition is based ...
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### Set of polynomials and dimension stability under addition

I consider the subset $\mathcal{U}\subset\mathbb{R}_{2}[x_1,x_2]\times \mathbb{R}_{2}[x_1,x_2]$ defined by $$\mathcal{U} = \{(x_1,x_2)\in[0,1]^2\mapsto (ax_1,b(1-x_1)x_2 : (a,b)\in\mathbb{R}^2\}$$ ...
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### Difficulty to prove that a set is semi algebraic

I consider for $i\in\{1,…,N\}$ the subset $X_i\subset\mathbb{R}^{n}$ that are convex and compact and semi algebraic. We denote by $X=\Pi_{i}^{N}X_i$ the Cartesian product of the $X_i$ which is still ...
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### Is a subgroup of $\operatorname{GL}(n,\mathbb{R})$ semialgebraic if and only if all its orbits are?

A subset $X \subseteq \mathbb{R}^n$ is called semialgebraic if it is of the form $$X = \bigcup_{finite} \bigcap_{finite} \{ x \in \mathbb{R}^n \colon f_{i,j}(x) \star 0 \}$$ where $\star$ represents ...
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### Does there exist some o-minimal structure such that a given strictly monotonic function $\mathbb{R} \to \mathbb{R}$ is definable?

Before I start a little disclaimer: I am a little bit new to the concept of o-minimal structures, so my apologies in advance if this question is a bit on the trivial side of things. If so, could you ...
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### Image of a function can't contain an open set

I am trying to prove that the image of $(x, y) \mapsto (x, xy, xye^{y})$ isn't a semianalytic set on $(0,0,0)$. In order to do so, I need to prove that its image can't contain an open set. How would I ...
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### Semi-algebraic neighbourhoods of a semi-algebraic set

Let $S\subset\mathbb{R}^n$ be a semi-algebraic set, and let $U\subset \mathbb{R}^n$ be an open neighbourhood of $S$ (non necessarily semi-algebraic). Is it true that there exists an open semi-...
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### Prove that infimum function is semi-algebraic by Tarski-Seidenberg theorem

I am reading the paper "A. D. Ioffe, An invitation to tame optimization, SIAM J. Optim., 19 (2009), 1894–1917" and stuck at Proposition 3.1. The author claims (in the language of semi-...
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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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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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 ...
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### Zero set of nested radicals

My question deals with a function on $\mathbb{R}^n$ that consists of nested radicals and polynomial functions. I'm not even sure how to properly formulate this question, i.e. precisely what class of ...
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### Sufficient conditions on a semialgebraic set for being a manifold

Let $A = \{x \in \mathbb{R}^n \ | \ \forall i \in \{1,...,m\}, P_i(x) \geq 0\}$ be a semialgebraic subset of $\mathbb{R}^n$ (for some $n,m \in \mathbb{N}^\ast$). Are there some sufficient conditions ...
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### Is there a rational homeomorphism $\mathbb{R}^2 \approx B_1(0,0)$ with rational inverse?

Let $B_1(0,0) \subseteq \mathbb{R}^2$ denote the open ball of radius $1$ about the origin. One of the first things you learn while studying (general) topology is that $\mathbb{R}^2 \approx B_1(0,0)$. ...
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### Proving that a certain region in $\mathbb{R}^2_{>0}$ has a bounded number of intervals when intersected with a line

Let $R$ be a subset of $\mathbb{R}^2_{>0}$ such that the following conditions hold: $$x < y < x(1 + \sqrt{1 + \frac{B}{A}})$$ $$y(Ay + Bx) \leq E$$ $$C \leq y$$ $$D \leq A y + B x$$ ...
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### Stability under operations for semi algebraic functions

Let $E$ and $F$ two semialgebraic sets. Let $\phi(\alpha,\theta): E\times F$ a bounded semi-algebraic function. How to prove that $\theta\mapsto \sup_\alpha\phi(\alpha,\theta)$ is still semi-...
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### Image of semialgebraic set is a semialgebraic set

Let $f: \mathbb{R}^n \to \mathbb{R}^m$, where $m\le n$, be a linear map that maps all points with rational coordinates to the points with rational coordinates. Prove that the $f(A)$ is semialgebraic ...
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### Is a square a semi-algebraic set?

Consider the square $[-1,1]^2$ in $\mathbb R^2$. Is this set semi-algebraic?
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### Given infinitely many linear inequalities, can we find an algebraic equation for the boundary?

Say we have infinitely many linear inequalities cutting out a set of points inside some $\mathbb{R}^n$. Is there any systematic way of translating these into an algebraic equation for the boundary of ...
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### Belonging to the same connected component of a semialgebraic set

Warm-up and main questions: Let $X \subset \mathbb{R}^n$ be a semialgebraic set and $x_1, x_2 \in X$. How can I effectively check if $x_1$ and $x_2$ are in the same connected component of $X$? Let \$f ...