# 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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### If a subset $X$ of a $C^k$ manifold $M$ is semialgebraic in the charts of $M$, is it Whitney stratifiable?

Let $M$ be a $C^k$ manifold for some $k\geq 1$ and $X$ be a subset of $M$. Assume that there is an atlas of charts $(\phi_\alpha, U_\alpha)_\alpha$ of $M$ such that in the coordinates of each of these ...
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### Feasibility of a polynomial system of equalities and inequalities

Consider a system of the form $f_i(x) = 0$ and $g_j(x) \ge 0$ ,where $f_i,i=1,\dots,r$ and $g_j,j=1,\dots,s$ are polynomials in real unknowns $x_i,i=1,\dots,n$ with rational coefficients. Is there a ...
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### 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 $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...
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### Tarski-Seidenberg for strict inequalities and bounded quantification

This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
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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-...
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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 ...
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### 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 ...
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