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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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(...
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Certificates of connectivity of basic semi-algebraic sets

Given real polynomials $p_1, \ldots, p_n \in {\mathbb R}[x_1, \ldots, x_d]$, consider the closed basic semi-algebraic set $S \subseteq {\mathbb R}^d$ given by $$S := \{x \in {\mathbb R}^d : p_i(x) \...
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6 votes
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Homotopy equivalence of stably equivalent semialgebraic sets

In his book [1], Richter-Gebert introduces a notion of stable equivalence for primary basic semialgebraic sets (subsets of $\mathbb{R}^n$ defined by a conjunction of polynomial equations and strict ...
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2 votes
0 answers
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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 ...
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What are the possible asymptotics of the measure of a parametrised semialgebraic set?

Consider a family of semialgebraic sets $S_t \subset \mathbb{R}^d$ ($t \in [0,1]$) of the form $$ S_t = \{ x \in \mathbb{R}^d \ : \ p_1(x,t) \geq 0,\ p_2(x,t) \geq 0, \dots, p_m(x,t) \geq 0 \} $$ ...
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6 votes
1 answer
99 views

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 ...
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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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6 votes
1 answer
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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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5 votes
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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 ...
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1 vote
2 answers
284 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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10 votes
2 answers
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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 ...
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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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11 votes
1 answer
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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 $f$ ...
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12 votes
2 answers
703 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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4 votes
1 answer
142 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 ...
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2 votes
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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 ...
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7 votes
1 answer
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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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3 votes
0 answers
74 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 = \...
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4 votes
2 answers
466 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 ...
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5 votes
1 answer
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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. ...
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6 votes
0 answers
240 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 ...
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0 votes
0 answers
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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. ...
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8 votes
1 answer
235 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,\...
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11 votes
2 answers
378 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 ...
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7 votes
0 answers
196 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 ...
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4 votes
1 answer
223 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?
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5 votes
1 answer
279 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{...
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4 votes
1 answer
175 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 ...
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1 vote
1 answer
205 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 \...
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