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 $f_1,\ldots,f_s\in \mathbb{R}_m[x_1,\ldots, x_n]$.

Is the dependence of $A(f_1,\cdots,f_s)\cap B$ on the polynomials $f_1,\cdots,f_s$ outer semicontinuous?

Are there known results like this in the theory of real semialgebraic sets?

A set-valued function $F:X\to \mathcal{P}(\mathbb{R}^n)$, that to each $x\in X$ associates a compact subset $F(x)\subset\mathbb{R}^n$, is called outer semicontinuous when for every $x_0\in X$ and $\delta>0$ there exists a neighborhood $U$ of $x_0$ such that $F(x)\subset N_\delta(F(x_0))$ for all $x\in U$.

$N_\delta(A)$ stands for a $\delta$-neighborhood of a set $A\subset \mathbb{R}^n$.

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    $\begingroup$ Can you define the concept of outer semicountinuity of a familly of sets? $\endgroup$ – Liviu Nicolaescu Sep 26 '15 at 22:58
  • $\begingroup$ Which topology do you consider on the space of polynomials? The number $s$ is probably fixed. What about the degrees? $\endgroup$ – Jochen Wengenroth Sep 30 '15 at 13:29
  • $\begingroup$ Sorry for the slopiness, I meant s fixed and the degrees of the polynomials bounded, so that the topology is clear. $\endgroup$ – p. duarte Oct 1 '15 at 10:54

This seems like it can't be true since the sets $A(f_1, \ldots, f_s)$ may not be compact, which violates the precondition for outer semicontinuous, but also makes the postcondition not hold.

E.g., consider $C = \{(x, y): xy-1 \ge 0\}$. It does not seem like we can force $A(f) \subseteq N_\delta(C)$ for all $f$ near $xy-1$. You will have to use some $f \in U$ containing an $x$ term, which will already make $A(f)$ very different from $A(xy-1)$.

  • $\begingroup$ Sorry, the compactness assumption was implicit in my question. I have reformulated it to make it explicit. $\endgroup$ – p. duarte Sep 30 '15 at 9:24

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