The ring of bounded polynomials on a semi-algebraic set (Q2841366)

The authors consider an affine \(\mathbb{R}\)-variety \(V\) with coordinate ring \(\mathbb{R}[V]\) together with a semi-algebraic set \(S\) contained in \(V(\mathbb{R})\), the set of \(\mathbb{R}\)-points. They study the subring \(B_V(S)\subseteq\mathbb{R}[V]\) whose elements are the functions that are bounded on \(S\). The properties of \(B_V(S)\) depend on the geometry of \(V\) and \(S\). There are some trivial situations, e.g., if \(V(\mathbb{R})\) or \(S\) is compact. Otherwise it is a difficult problem to analyze \(B_V(S)\). The authors' approach is to use completions of the variety \(V\). They introduce the notion of an \(S\)-compatible completion and prove:NEWLINENEWLINE If \(V\) is normal and if \(X\) is an \(S\)-compatible completion, then there is an open subvariety \(U\subseteq X\) such that \(V\subseteq U\) and the canonical homomorphism \(O_X(U)\to\mathbb{R}[V]\) is an isomorphism onto \(B_V(S)\).NEWLINENEWLINE This result raises the question whether there is an \(S\)-compatible completion for all \(V\) and S\(.\) The general answer is negative, of course. But there are some positive answers, in particular in the case that \(V\) is a surface. Finally, the authors consider the question whether, or when, the ring \(B_V(S)\) is finitely generated.