17
votes
Polynomials that preserve nonnegativity
Such a polynomial may be represented as $q^2 +xr^2 $ (proof: this is true for all irreducible divisors which have degree 1 or 2, and the set of polynomials of this kind is a multiplicative monoid by ...
- 109k
12
votes
Accepted
General Tarski-Seidenberg Theorem
The most abstract version of the Tarski-Seidenberg theorem I know of is the following
Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map
$$f^*:\...
- 16.5k
10
votes
Accepted
Projections of compact real algebraic sets
The answer is no. (Although the previous example I gave was bad.)
Let $C$ be the curve $y^2 = x^2 (x-1)(2-x)$, so $C$ has a smooth component with $1 \leq x \leq 2$, and also a node at $(0,0)$. Let $M$ ...
- 156k
7
votes
Accepted
First order decidability of limit of gradient flow?
Overview: The boundaries of the basins of attraction are lower dimensional stable manifolds. In two dimensions, they are the arcs flowing from repelling fixed points to saddle points. I expect that ...
- 156k
6
votes
Is a spectrahedron's boundary almost always "smooth"?
There are many interesting examples where there are singularities on the border. E.g. spectrahedra describing decompositions of polynomials as sums of squares (s.o.s.) of polynomials have ...
- 14k
4
votes
Semialgebraic sets containing irrational power functions
The answer to this question is negative:
Jut take $\alpha$ such that $2^\alpha=3$. Taking into account that $2^n\ne 3^m$ for any natural numbers $n,m$, we can prove that the number $\alpha=\log_23$ ...
- 41.8k
3
votes
Lower convex envelope of polynomial functions
The answer is yes, but not in the most exciting way. The idea is to write the envelope in the first-order logic and then kill the problem with quantifier elimination for the theory of real-closed ...
- 6,476
3
votes
Accepted
Do convex closed semialgebraic hyperplane cross-sections imply semi-algebraicity?
Here I record an answer based on very kind comments above (and offline).
The construction starts from a closed ball $B\subset\mathbb{R}^3$ and an infinite sequence $\mathcal{L}:=\{L^+_k\}$ of open ...
- 14k
3
votes
Is a spectrahedron's boundary almost always "smooth"?
Let $\mathrm{Sym}^2(\mathbb{R}^n)$ be the $\binom{n+1}{2}$ dimensional vector space of $n \times n$ symmetric matrices and let $P \subset \mathrm{Sym}^2(\mathbb{R}^n)$ be the cone of positive ...
- 156k
2
votes
On strict positivity and Schmüdgen's Positivstellensatz
This answer may be a bit late, but every homogeneous polynomial which is not a sum of squares of homogeneous polynomials (like $Z^6 + X^4 Y^2 + X^2 Z^4 - 3(XYZ)^2$), when restricted to any ball of ...
- 21
2
votes
Accepted
Constructing M-curves à la Hilbert
As Zach Teitler hinted, I shouldn't have checked it visually. In fact, if you make the construction correctly, it is quite easy to show that you have 8 real intersection with one of the ellipse and ...
- 2,275
2
votes
The set of polytopes with given $f$-vector
It is semi-algebraic. Let $x$ be an $f_0$-tuple of points in $R^n$.
Take any subset of $n$ points from $x$, and find a hyperplane through
these points. This involves only rational operations on the ...
- 91.8k
2
votes
Accepted
Is the polar dual of a semi-algebraic convex body also semi-algebraic?
The comment of Robert brought me onto the right track.
Say $I=\{1,...,m\}$, then
\begin{align}
\Bbb R^n\setminus C^\circ
&= \{\,y\in\Bbb R^n\mid \exists x\in C\colon\langle x,y\rangle >1\,\}
\\...
- 13.6k
2
votes
Bounding distance to an intersection of semialgebraic sets
You also need convexity for this to be true. Otherwise, a counterexample, e.g.:
$$\DeclareMathOperator{\dist}{dist}
\begin{split}
P &= \{xy<1\},\\
Q &= \{(1-x)y<1\},
\end{split} (x,y)\...
- 21
1
vote
Are there polytopes with precisely two realizations?
Yes, and there are infinitely many of them. If you have trouble visualizing Gale duality, the easiest thing may be to instead look at the way Lawrence lifts go step by step (as is explained in our ...
- 732
1
vote
Accepted
Dimension and cardinality of fibers over the real numbers
The answer to (1) is yes and holds more generally for any o-minimal structure on $\mathbb{R}$. If $f:X \to Y$ is any definable map between definable sets in an o-minimal structure, then there is a ...
- 2,971
1
vote
Accepted
Homotopy equivalence of stably equivalent semialgebraic sets
The quick answer to my question is: Yes, the space described in the question is a real counterexamle. It can even be "homogenized" to $W = \{\, (v,w,w') \in \mathbb{R}^3 : w' > 0, \, v(vw ...
- 136
1
vote
Accepted
Solutions to a system of homogeneous equations (inequalities)
I cannot say much about $\geq0$. For $=0$ one can prove the following statement: if $r\leq n-1$ then then there is an $a\in \mathbf R^n\setminus\{0\}$ such that $f_i(a)=0$ for all $i$.
Indeed, each ...
- 1,424
1
vote
Complex semi-algebraic sets
I will jot down some stray (and easy) thoughts, in an effort to engage the question and see whether some aspects of it can be made more precise.
Going out on a limb, I suppose a baseline assumption ...
1
vote
Complex semi-algebraic sets
Here I unroll the argument in [MR2399570] showing that $\mathbb{L}^n=\mathbf{T}^{-1}(S)$ for a semialgebraic set $S$, to show that at worst [loc.cit.] is a bit too brief, but no more that that.
Given ...
- 14k
1
vote
Accepted
Polynomials that preserve nonnegativity
The result is known as the Pólya-Szegö theorem:
Note that $\Sigma = \Sigma_n := \{ p \in \mathbb{R}[x] \mid p~\text{SOS}\}$, in which 'SOS' stands for sum of squares.
A reference for the result: ...
- 1,719
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