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Questions tagged [semialgebraic-geometry]

Semialgebraic geometry is the study of semialgebraic sets.

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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 ...
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45 views

Which topological property am I looking for?

I just started reading about topology and semi-algebraic sets, so please forgive me if this is a naive question: Given a bounded and closed semi-algebraic set, which topological property guarantees ...
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Estimate “exponent” of a Radical ideal in Ring of continuous function and semi-algebraic function

We know definition of radical ideal. Let $I$ is a ideal of a ring X then radical ideal of $I$ noted by $\sqrt I $ = $\sqrt I = \left\{ {x\left| {{x^l} \in I,l\text{ is a some natural number}} \right.}...
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Whether a Convex function is a semi-algebraic function, a convex set is a semi-algebraic set?

Definition: A semi-algebraic set is a subset described by equality and inequality of polynomials. For example: $\left\lbrace x \;|\; f(x)>0, \;g(x)=0, \ldots \right\rbrace$. Graph $(x,f(x))$ of a ...
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3answers
408 views

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 ...
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50 views

Duality gap in polynomial optimization problem Lasserre relaxation

Consider a polynomial optimization problem of the following type \begin{equation} \begin{array}{cl} \text{maximize} & p(\mathbf{x}) \\ \text{subject to} & \mathbf{x} \in K \\ \end{array} \...
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61 views

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 $\...
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2answers
114 views

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 ...
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0answers
55 views

Disjoint Union Basic-Closed Semialgebraic Sets

A basic-closed semialgebraic set in $\mathbb{R}^{n}$ is defined as $$ M=\lbrace x \in \mathbb{R}^{n} \mid f_{1}(x) \geq 0, \ldots, f_{m}(x) \geq 0 \rbrace$$ for some polynomials $f_{i}$ in $n$ ...
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1answer
99 views

Lebesgue measure of a semi-algebraic set

Let $A$ be a semi-algebraic subset of $\mathbb{R}^n$ that is semi-algebraically homeomorphic to $(0,1)^k$ with $k<n$. I would require a result stating that $\mathcal{L}^n(A)=0$ where $\mathcal{L}^...
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1answer
66 views

What geometry (if any) studies sets defined by infinitely many polynomial inequalities?

Question: Semi-algebraic geometry studies the solution sets of finitely many polynomial inequalities (in $\mathbb{R}^n$). What field (if any) could be considered the study of the solution sets of ...
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74 views

Area of semialgebraic set

Consider the following system of polynomial inequalities with variables $q_1,\ q_2$. \begin{align*} a_1 \leq q_1 + q_2 \leq a_2\\ b_1 \leq q_1^2 + q_2^2 \leq b_2\\ c_1 \leq q_1^3 + q_2^3 \leq c_2 \\ ...
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1answer
82 views

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<...
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0answers
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Semialgebraic sets with irrational exponents

A semialgebraic set is defined by finite unions and complements of inequalities of the form $g(x)\ge 0$ where $g$ is a multivariate polynomial with integer coefficients. My question considers the ...
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2answers
104 views

The difference between two algebraic sets is not closed?

Let $A$ and $B$ be two non-empty algebraic subsets of $\mathbb{C}^n$ such that $B$ is strictly contained in $A$. I am trying to show that the difference $A-B$, which is a semi-algebraic set, is not ...
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48 views

Path Minimizing Distances on Semi-Algebraic Surfaces

Let $M$ be a $k$-dimensional semi-algebraic manifold embedded in $\mathbb{R}^n$. Assume that $M$ is diffeomorphic to $\mathbb{R}^k$. We are interested in the Euclidean path minimizing distance ...
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1answer
90 views

Finding extrema of sublevel set of the Hénon-Heiles potential function

The inequality $$\frac{1}{2}(x^2+y^2)+x^2y-\frac{1}{3}y^3 \lt C$$ produces a somewhat triangular shape near the origin for $C \le \frac 16$, where $C$ is a constant. Example when $C = 1/10$: I am ...
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3answers
185 views

Is the set $S = \{x \in [0, \infty)^2 \mid x_1 x_2^2 \leq 1\}$ convex?

We have the function $\displaystyle{y=f(x_1, x_2)=x_1\cdot x_2^2}$ and the set $S=\{x\in [0, \infty)^2 \mid f(x_1, x_2)\leq 1\}$. I want to check if the set is convex. $$$$ Let $x=(x_1, x_2) , y=...
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2answers
540 views

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 ...
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1answer
74 views

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 $\...
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2answers
1k views

How to plot the PSD cone in MATLAB

Does anybody know how I can plot in MATLAB the cone of positive semidefinite matrices as shown in the figure below? Thanks. PSD cone
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1answer
221 views

A spectrahedron is a quadric cone when matrices in LMI are in $\mathbb{R}^{2\times 2}$

I am watching a lecture (just at the beginning around 0:50-0:57). The note says when $n=2$, $\mathcal{S}$ is a quadric cone; however, it seems that the professor says "a quadratic cone". On ...
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0answers
59 views

Methods to compute volume of a semi-algebraic region

I am interested in a specific question involving the computation of the volume (perhaps area is more appropriate?) of a semi-algebraic region defined by a ternary quadratic form and a ternary cubic ...