The ring of bounded polynomials on a semi-algebraic set

Abstract:

Let $V$ be a normal affine $\mathbb {R}$-variety, and let $S$ be a semi-algebraic subset of $V(\mathbb {R})$ which is Zariski dense in $V$. We study the subring $B_V (S)$ of $\mathbb {R}[V]$ consisting of the polynomials that are bounded on $S$. We introduce the notion of $S$-compatible completions of $V$, and we prove the existence of such completions when $\dim (V)\le 2$ or $S=V(\mathbb {R})$. An $S$-compatible completion $X$ of $V$ yields a ring isomorphism $\mathscr {O}_U(U)\overset {\sim }{\to } B_V(S)$ for some (concretely specified) open subvariety $U\supset V$ of $X$. We prove that $B_V(S)$ is a finitely generated $\mathbb {R}$-algebra if $\dim (V)\le 2$ and $S$ is open, and we show that this result becomes false in general when $\dim (V)\ge 3$.