All Questions
Tagged with semialgebraic-geometry ag.algebraic-geometry
21 questions
1
vote
1
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104
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Dimension and cardinality of fibers over the real numbers
If $X \subseteq \mathbb{C}^n$ and $Y \subseteq \mathbb{C}^m$ are irreducible affine varieties, and $f : X \to Y$ is a dominant polynomial map, then we know that (every irreducible component of) the ...
- 155
4
votes
0
answers
119
views
Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$
The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...
- 13.6k
2
votes
0
answers
181
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An application of 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-...
- 21
9
votes
0
answers
289
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Computer algebra tools for finding real dimension of an algebraic variety
I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set?
The CAD-based ...
- 175
2
votes
1
answer
108
views
Is the polar dual of a semi-algebraic convex body also semi-algebraic?
Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as
$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$
with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a ...
- 13.6k
9
votes
1
answer
265
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Projections of compact real algebraic sets
Suppose that $M$ is a compact, real
algebraic subset of $\mathbb R^n$ and $f:\mathbb R^n \to \mathbb R^m$ is the projection to the first $m$ coordinates. If $f$ maps $M$ bijectively unto its image $f(...
- 2,934
2
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0
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194
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Examples of semi-abelian schemes over a curve
Let $C$ be a nice curve, i.e. $C$ is a smooth, projective, geometrically integral scheme of dimension $1$ over a field $k$. For example, (assuming the characteristic of $k$ is neither 2 or 3) an ...
- 452
6
votes
1
answer
257
views
Constructing M-curves à la Hilbert
I have been reading some text about Harnack's theorem. The theorem basically says that for degree $d$, the maximal number of connected components in the real (projective) plane of a plane curve with ...
- 2,275
7
votes
1
answer
749
views
General Tarski-Seidenberg Theorem
The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of ...
- 3,599
1
vote
0
answers
35
views
Convex combination of semi-algebraic sets
Suppose we are given two semi-algebraic sets $S_1$ and $S_2$. Define
$$
S=\{s=ps_1+(1-p)s_2: s_1\in S_1, s_2\in S_2, 0\leq p\leq 1.\}
$$
$S$ is semi-algebraic.
Can we bound the degree of $S$?
If we ...
- 1,503
10
votes
2
answers
224
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 ...
- 3,079
4
votes
1
answer
324
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 ...
- 2,837
13
votes
1
answer
340
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 $f$ ...
- 505
12
votes
2
answers
742
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 $p(x) \ge 0$ ...
- 1,719
4
votes
1
answer
155
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 ...
- 203
2
votes
0
answers
117
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 ...
- 51
5
votes
1
answer
169
views
Connectedness of semialgebraic sets via CAD
I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...
- 61
11
votes
2
answers
418
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 ...
- 715
7
votes
0
answers
214
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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 ...
- 101
5
votes
1
answer
298
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
1
answer
183
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 ...
- 41