Mathematics

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.

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.

75 questions
21
votes
2 answers

What is the volume of the $3$-dimensional elliptope?

My question Compute the following double integral analytically $$\int_{-1}^1 \int_{-1}^1 2 \sqrt{x^2 y^2 - x^2 - y^2 + 1} \,\, \mathrm{d} x \mathrm{d} y$$ Background The $3$ -dimensional elliptope is the spectrahedron defined as follows $$\mathcal…
Rodrigo de Azevedo
  • 23,788
10
votes
3 answers

Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$ $$ P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij}, $$ I would like to find a matrix $P$ which satisfies…
Alex Shtoff
  • 208
9
votes
2 answers

Why is the set $\{ (x,y,z) \geq 0: 3(xy+yz+xz) +1 \geq 2(x+y+z), x+y+z \leq 1\}$ convex?

While trying to solve some problems, the following set $$ S:= \{ (x,y,z): x,y,z \geq0 \} \cap \{ (x,y,z): 3(xy+yz+xz) +1 \geq 2(x+y+z) \} \cap \{ (x,y,z): x+y+z \leq 1 \} $$ showed up in my calculations and I needed to prove that this set is indeed…
Psychomath
  • 2,065
6
votes
1 answer

Structures strictly between $\{\mathbb{R},+,<\}$ and $\{\mathbb{R},+,\cdot,<\}$.

Let $\mathcal{R}_0=\{\mathbb{R},+,<\}$ be the order divisible abelian group of reals and $\mathcal{R}=\{\mathbb{R},+,\cdot,<\}$ be the real closed field of reals. Both $\mathcal{R}_0$ and $\mathcal{R}$ are o-minimal structures. Let…
Anguepa
  • 3,447
5
votes
1 answer

Analogue of semialgebraic sets over complex numbers

Real semialgebraic sets are sets definable in the language of the reals: $(\mathbb{R},0,1,+,\cdot)$, which has as a definitional extension $(\mathbb{R},0,1,+,\cdot,\leq)$ by the useful fact that $a< b$ iff there exists $t\in\mathbb{R}$ such that…
KReiser
  • 76,664
4
votes
1 answer

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 any of the symbols $=, \leq , \geq, <, >$ and…
Strichcoder
  • 2,025
4
votes
1 answer

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 $(\Bbb R,<)$ of real numbers. Show that for a…
Alessandro Codenotti
  • 13,145
4
votes
1 answer

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}^p\bigcap_{i=1}^q\{x\in\mathbb{…
RamenHunter
  • 93
4
votes
0 answers

Lie group and algebraic group with isomorphic Lie algebra

This might be very silly but I have been struggling with this for a while now... Consider two connected linear Lie groups $G\leq GL_n(\mathbb{R})$ and $H\leq GL_m(\mathbb{R})$ with Lie algebras $\mathfrak{g} \leq \mathfrak{gl}_n(\mathbb{R})$ and…
WrabbitW
  • 597
4
votes
2 answers

Every convex polyhedron is a spectrahedron

I'm trying to show that convex polyhedra are special cases of spectrahedra. This was left as an exercise to the reader in a convex optimization text that I'm reading. I'm not sure how standard the definitions and notation in this book are, so I'll…
zxmkn
  • 1,640
3
votes
0 answers

Areas of four triangles in a cyclic quadrilateral

Let $P_1,P_2,P_3,P_4$ be four distinct, cyclically ordered points on a circle. Write $S_{ijk}$ for the area of $\triangle P_iP_jP_k$ . Then $(S_{123},S_{124},S_{134},S_{234})$ lies in a 3-dimensional semialgebraic region contained in the…
hbghlyj
  • 6,799
3
votes
0 answers

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 $\mathbb{C}$ because it isn't an ordered field, but we do…
Harry Wilson
  • 1,124
3
votes
0 answers

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 defined on the compact semialgebraic set $V \subset…
André
  • 181
3
votes
0 answers

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 nonempty interior in this topology? It seems like a…
subrosar
  • 5,310
3
votes
2 answers

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 which contains the graph of the exponential function…
Mario G
  • 444
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