The ring of bounded polynomials on a semi-algebraic set
Daniel Plaumann, Claus Scheiderer · Transactions of the American Mathematical Society · 2012
Let V V be a normal affine R \mathbb {R} -variety, and let S S be a semi-algebraic subset of V ( R ) V(\mathbb {R}) which is Zariski dense in V V . We study the subring B V ( S ) B_V (S) of R [ V ] \mathbb {R}[V] consisting of the polynomials that are bounded on S S . We introduce the notion of S S -compatible completions of V V , and we prove the existence of such completions when dim ( V ) ≤ 2 \dim (V)\le 2 or S = V ( R ) S=V(\mathbb {R}) . An S S -compatible completion X X of V V