# 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.

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### Lower convex envelope of polynomial functions

Let $P\in{\mathbb R}[X]$ be a polynomial and $[a,b]$ be a bounded interval. Of course, the graph of $P$ is an algebraic set. I am interested in the lower convex envelope $\bar P$ of $P|_{[a,b]}$. It ...
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### 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 ...
70 views

### Representations in Archimedean quadratic modules

Let $\mathbb R [X] = \mathbb R [X_1,\dots,X_n]$ and $\Sigma[X] = \big\{ \, f \in \mathbb R[X] \mid \exists r \in \mathbb N, \ g_i \in \mathbb R[X] \colon f = g_1^2 + \dots + g_r^2 \,\big\}$ denote ...
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### 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 ...
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### On strict positivity and Schmüdgen's Positivstellensatz

Schmüdgen's Positivstellensatz requires the polynomial to be strictly positive on a semialgebraic set. While trying to understand it, I am just wondering if the strictly positive condition can be ...
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### 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 ...
242 views

### Complex semi-algebraic sets

Definition. A subset $S$ of $\mathbb{R}^n$ is called semi-algebraic if $S$ is a finite Boolean combination of sets of the form $\{ x \in \mathbb{R}^n \mid p(x) \ge 0\}$, where $p \in \mathbb{R}[x]$. ...
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### 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 ...
201 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 ...
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### 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 $f$ ...
679 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 $p(x) \ge 0$ ...
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### 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 ...
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### 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 ...
218 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 ...
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### 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 ...
189 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 ...
169 views

### Compactness of a semi algebraic set

Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?
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, $l \in \mathbb{... 1answer 173 views ### Ultrafilters on the set of semialgebraic subsets of R^2 Let$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 ...
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 \...