Abstract. Let V be a normal affine R-variety, and let S be a semi-algebraic subset of V (R) which is Zariski dense in V. We study the subring BV (S) of 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) ≤ 2orS = V (R). An S-compatible completion X of V yields a ring isomorphism OU (U) ∼ → BV (S) for some (concretely specified) open subvariety U ⊃ V of X. We prove that BV (S) is a finitely generated R-algebra if dim(V) ≤ 2andS is open, and we show that this result becomes false in general when dim(V) ≥ 3
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