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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 ...
G2MWF's user avatar
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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 ...
G2MWF's user avatar
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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 $$ \mathcal{U} = \{(x_1,x_2)\in[0,1]^2\mapsto (ax_1,b(1-x_1)x_2 : (a,b)\in\mathbb{R}^2\} $$ ...
G2MWF's user avatar
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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 ...
G2MWF's user avatar
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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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Terminology for Complex Algebraic Geometry with Complex Conjugation

Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities. This doesn't make sense over $\mathbb{C}...
Harry Wilson's user avatar
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Finding the closest positive definite matrix

Let $L$ be the symbolic $n \times n$ lower triangular matrix and $S$ a real symmetric $n \times n$ positive semidefinite matrix. Fix a real symmetric $n \times n$ positive definite matrix $A$ with the ...
12345's user avatar
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1 answer
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Finding the image of a semialgebraic set under a linear projection

I have a semi-algebraic set defined by the equality constraints $\{f_i (x_0, \dots, x_n) = 0\}_i$ and inequality constraints $\{g_j(x_0, \dots, x_n) \leq 0\}_j$ for real polynomials $f_i, g_j \in \Bbb ...
Waylander's user avatar
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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 ...
Sliem el Ela's user avatar
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Question regarding definition of semialgebraic set

I read from notes that a semi-algebraic set in $\mathbb{R}^n$ is a set that can be given by finitely many polynomial equalities and inequalities. It is a finite union of sets of the form $\{x \in \...
Chang Henry's user avatar
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2 answers
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Is volume of semialgebraic sets definable over the language of ordered fields?

Let $ \mathcal{L}=\{0,1,+,\cdot,\le\} $ be the language of ordered fields and consider the theory of $\mathbb{R} $ in this language (i.e., the theory of real closed fields). Suppose $ \varphi(x_1,\...
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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 ...
user480840's user avatar
2 votes
1 answer
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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-...
Antonio 's user avatar
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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-...
Dat Ba Tran's user avatar
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1 answer
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Nonnegativity of sparse polynomial with just one negative coefficient

Consider the polynomial $$ P(x,y)=15 \times 10^{15} (x^4+y^4)+2 \times 9999^2 \times 10^{12} xy(x+y-3)+20001 \times 9999^4 $$ One can show that $P(x,y)\geq 0$ when $x,y\geq 0$ (proof is below). Since $...
Ewan Delanoy's user avatar
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Triangulation Theorem for semialgebraic maps

Benedetti and Risler's "Real algebraic and Semi-algebraic sets" book on Semialgebraic Geometry has the following theorem: Theorem 2.6.14 Let $f:V \to Y$ be a continuous semialgebraic mapping ...
André's user avatar
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The derivative of a semialgebraic map is semialgebraic

Coste's notes on semialgebraic geometry have the question: If $f:U \to \mathbb{R}$ is semialgebraic, with $U$ an open semialgebraic set, then each partial derivative $\dfrac{\partial f}{\partial x_{i}...
André's user avatar
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1 answer
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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 ...
Matthew Kahle's user avatar
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2 answers
292 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 ...
André's user avatar
  • 159
0 votes
1 answer
169 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 ...
André's user avatar
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1 vote
1 answer
158 views

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 ...
subrosar's user avatar
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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 ...
André's user avatar
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2 votes
1 answer
167 views

Semialgebraic Morse-Sard Theorem - why are the critical points of a semi-algebraic map again semi-algebraic?

2.5.12 Exercise (Semi-algebraic Morse-Sard theorem) Let $f:M\to N$ be a $C^\infty$ semi-algebraic map between semi-algebraic submanifolds of $\Bbb R^n$ and $\Bbb R^m$, respectively. Set $$C = \{x\in M;...
André's user avatar
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3 votes
2 answers
218 views

Closed semialgebraic subset of $\mathbb{R}^2$

I'm trying to solve the following problem from exercise 2.13 of Michel Coste's An introduction to Semialgebraic Geometry (October 2002) [PDF]. Let $S$ be a closed semialgebraic subset of the plane ...
Mario G's user avatar
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1 vote
1 answer
160 views

Why is the image of a subanalytic set through a proper analytic map subanalytic?

Recall the following definitions, where $M$ and $N$ are real analytic manifolds: A subset $A\subseteq M$ is called semianalytic if for every $m\in M$ there is an open $U_m\subseteq M$ such that $U_m\...
TopologicalDynamitard's user avatar
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Are the irreducible components of a definable closed analytic set themselves definable?

Recall that a structure on $\Bbb R$ is defined to be a sequence $\mathcal S=(S_n)_{n<\omega}$ such that for each $n\geq 0$ the following properties hold: $S_n$ is a boolean algebra of subsets of $\...
TopologicalDynamitard's user avatar
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123 views

Why is the singular locus of a definable closed analytic set definable?

Let $X\subseteq\Bbb C^n$ be a closed analytic subset, meaning that for every $x\in X$ there is an open subset $U_x\subseteq\Bbb C^n$ and a finite family of holomorphic functions $f_1,\ldots,f_k\colon ...
TopologicalDynamitard's user avatar
2 votes
1 answer
105 views

Why is $\exp(x)$ not definable in any interval in $\Bbb R_{\mathrm{an}}$?

According to these notes by Yilong Zhang the function $x\mapsto e^x$ is not definable in any interval in the $o$-minimal structure $\Bbb R_{\mathrm{an}}$, where the latter is defined to be the ...
TopologicalDynamitard's user avatar
1 vote
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59 views

Positive codimension of locus of singular points in semi-algebraic varieties

I am reading the paper A geometric proof of the existence of Whitney stratifications. Theorem 1 states Theorem 1. For any semivariety $V$ in $\mathbb{R}^m$ (or $\mathbb{C}^m$) there is an a- (resp. ...
jaogye's user avatar
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4 votes
1 answer
233 views

A definable set is definably connected iff it is connected?

I'm trying to solve exercise 2.19.7 from Chapter 3 of Van Den Dries's Tame Topology and O-minimal Structures, which is the following: Suppose $\mathcal S$ is an o-minimal structure on the ordered set $...
Alessandro Codenotti's user avatar
2 votes
2 answers
256 views

Detect if two elliptic cones overlap

Suppose I have two elliptic cones, both of whose vertices are at the same point. Do the interiors of these cones intersect? I'm working in normal 3-dimensional Euclidean space. An elliptic cone can ...
Dave's user avatar
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1 answer
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Can every semi-algebraic set be written as a finite union of disjoint Nash submanifolds?

Can we write every semi-algebraic set $A \subset \mathbb{R}^m$ as the disjoint union of finitely many Nash submanifolds of $\mathbb{R}^m$? By Nash submanifolds are meant those manifolds that are ...
b.omega's user avatar
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1 answer
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Semi-algebraic sets

The Astroid is a semi-algebraic curve. Since the points $(x,y)$ of the curve fulfill the equation $x^{2/3} + y^{2/3} = 1 $, and since they are points of said polynomial it is semi-algebraic. But my ...
user avatar
3 votes
1 answer
117 views

If $A\subset B$ are definable sets, and $A$ is open in $B$, then there exists a definable open $U$ with $U\cap B=A$

This is Lemma 3.4, chapter 1 in the book "Tame Topology and O-minimal Structures." Consider an o-minimal structure $\mathcal{S}=(\mathcal{S}_{n})$ on a dense linearly ordered nonempty set ...
cbyh's user avatar
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1 answer
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Density of the class of "semialgebraic" sets dense on that one of compact sets

Let $A \subseteq \mathbb R^n$ be a compact set and fix $\epsilon >0.$ Does there exists $m \in \mathbb N$ and polynomials $p_1,\ldots,p_m \in \mathbb R[x]$ such that the set $$S := \left\{x \in \...
John D's user avatar
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2 votes
1 answer
89 views

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 ...
MR_Q's user avatar
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0 answers
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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 ...
deeppinkwater's user avatar
2 votes
1 answer
58 views

When is a set $\{\mathbf x\in\Bbb R^n\mid p(\mathbf x)\le 0\}$ convex?

Given a multivariate polynomial $p(\mathbf x)=p(x_1,...,x_n)$ for $\mathbf x\in\Bbb R^n$. Are there some easy conditions to be set on (the coefficients of) $p$ to ensure that $$C_p:=\{(x_1,...,x_n)\...
M. Rumpy's user avatar
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4 votes
1 answer
838 views

Is the exponential function semi-algebraic?

Recall the following definitions: We say a set $E\subseteq\mathbb{R}^n$ is semi-algebraic if there exist real polynomials $g_{ij},h_{ij}:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $$E=\bigcup_{j=1}...
RamenHunter's user avatar
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1 answer
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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 $\mathbb{R}^2 \approx B_1(0,0)$. ...
asdf33313's user avatar
1 vote
1 answer
51 views

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: $$ x < y < x(1 + \sqrt{1 + \frac{B}{A}}) $$ $$ y(Ay + Bx) \leq E $$ $$ C \leq y $$ $$ D \leq A y + B x $$ ...
Johnny T.'s user avatar
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1 vote
1 answer
211 views

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-...
peanpean's user avatar
2 votes
1 answer
334 views

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 ...
Ronald S Merritt's user avatar
2 votes
2 answers
145 views

Is a square a semi-algebraic set?

Consider the square $[-1,1]^2$ in $\mathbb R^2$. Is this set semi-algebraic?
user avatar
1 vote
1 answer
71 views

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 ...
koobtseej's user avatar
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0 answers
144 views

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 ...
hakunamatata's user avatar