Let $A \subseteq \mathbb R^n$ be a compact set and fix $\epsilon >0.$ Does there exists $m \in \mathbb N$ and polynomials $p_1,\ldots,p_m \in \mathbb R[x]$ such that the set
$$S := \left\{x \in \mathbb R^n \mid p_i(x)\leq 0, i = 1,\ldots,m\right\}$$ is compact and satisfies $d_H(S,A)\leq \epsilon$? Here, $d_H$ denotes the Hausdorff distance between compact sets. Less idealy, I would be interested also on sets $S'$ satisfying the property, that are the finite union of sets with the structure of $S$ above.
I have no knowledge of the concepts and methods in algebraic geometry, but I believe this should be a well studied (possibly solved) problem in the field. References are also welcome.
Sure, because finite sets are dense with respect to the Hausdorff distance (cover $A$ by $\epsilon$-balls, pick a finite subcover by compactness, take the center of these balls). This even gives you that algebraic sets are dense with respect to Hausdorff distance.