Questions tagged [semialgebraic-geometry]
Semialgebraic geometry is the study of inequalities expressible in algebraic functions in one or several variables, usually over the real numbers or some field with similar properties.
10
votes
2answers
155 views
The set of polytopes with given $f$-vector
Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
3
votes
1answer
125 views
Solutions to a system of homogeneous equations (inequalities)
Let $f_1,\ldots,f_r \in \mathbb{R}[x_1,\ldots,x_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$.
For which values of $r,n,d$ there exists a real ...
11
votes
1answer
250 views
First order decidability of limit of gradient flow?
Let $f: \mathbb{R}^n\to\mathbb{R}$ be a polynomial function, and let $p$ be a critical point. Consider the ascending manifold $A_p$ consisting of all points whose limit under the gradient flow of $f$ ...
11
votes
2answers
558 views
Polynomials that preserve nonnegativity
A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be positive on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called nonnegative if $p(x) \ge 0$ ...
4
votes
1answer
122 views
Semialgebraic sets containing irrational power functions
Let $\alpha$ be an irrational number, and consider the set $A=\{(x,x^\alpha),x\ge 0\}\subseteq \mathbb{R}^2$, which is the graph of the function $f(x)=x^\alpha$.
I'm trying to prove/disprove the ...
2
votes
0answers
102 views
Classification of $n$-dimensional Nash-submanifolds of $\mathbb{R}^n$
Let $M,N\subset \mathbb{R}^n$ be two open semi-algebraic subsets, and assume that $M$ and $N$ are $C^\infty$ diffeomorphic, i.e. isomorphic as smooth submanifolds of $\mathbb{R}^n$. Does this imply ...
7
votes
1answer
202 views
Do convex closed semialgebraic hyperplane cross-sections imply semi-algebraicity?
Let $S\subset\mathbb{R}^n$, with $n\geq 3$, such that for any hyperplane $L$ one has $L\cap S$ closed, semialgebraic, and convex. Is it true that $S$ itself is semialgebraic?
A colleague explained to ...
2
votes
0answers
65 views
Constraints on cone of semialgebraic set
Let $S$ be a (basic) semialgebraic set, for which we know the constraints (i.e. we know the polynomial inequalities and equalities which define the set). Is it necessarily the case that the cone $C = \...
4
votes
2answers
269 views
Is a spectrahedron's boundary almost always “smooth”?
A spectrahedron is a convex set defined by a linear matrix inequality (LMI).
Is the boundary of such a set almost always smooth?
By "smooth" I mean that it admits a tangent hyperplane at any point ...
5
votes
0answers
187 views
Where do I read about semi-algebraic/analytic sets?
What's a good introduction to semi-algebraic/semi-analytic sets?
I'm coming from algebraic geometry, so ideally I'd prefer material that has a similar point of view and highlights analogies.
I've ...
0
votes
0answers
96 views
Cross sections of semialgebric sets
Let $n$ be bigger than two, and let $A$ be a subset of the $n$-dimensional Euclidean space.
Suppose that the intersection of $A$ with any $(n-1)$-dimensional affine hyperplane is semialgebraic.
...
8
votes
1answer
217 views
Do semialgebraic sets depend outer semicontinuously on their defining polynomials?
Consider a compact (semialgebraic) ball $B\subset \mathbb{R}^n$ and a semialgebraic set $A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n$ defined through some representation in terms of polynomials $f_1,\...
11
votes
2answers
251 views
Convexity of a certain sublevel set
Consider the polynomial of degree $4$ in variable $r$
$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$
The discriminant of this polynomial in $r$ is the following expression
(obtained using ...
7
votes
0answers
178 views
Moduli in semialgebraic geometry
Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry?
The sort of thing I am imagining is an object in a category of semischemes:
Ordinary schemes ...
5
votes
1answer
257 views
Convexity of a specific semialgebraic set
I have an engineering problem that may be solved using semidefinite programming. I would like to know whether a given set is convex.
Let $m \in \mathbb{R}^+$ be a positive real scalar, $l \in \mathbb{...
4
votes
1answer
167 views
Ultrafilters on the set of semialgebraic subsets of R^2
Let $R$ be a real closed field and $f: R \to R$ a map. Then let $\textrm{F}(f)$ be the set of semialgebraic subsets of $R^2$, which contain $(t,f(t))$ for all $0< t< \epsilon$ for some $\epsilon ...
1
vote
1answer
178 views
Is the closure of a semialgebraic set mod 1 also semialgebraic?
Let $p:\mathbb{R}^n\to[0,1)^n$ be the map defined by $p(x_1,\ldots,x_n)=(\{x_1\},\ldots,\{x_n\})$, where $\{\cdot\}$ is the fractional part operator. Experimentation suggests that if $S \subseteq \...
