# 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.

17 questions
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...