The Ring of Bounded Polynomials on a Semi-algebraic Set
نویسنده
چکیده
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 ) ≤ 2 or S = V (R). An S-compatible completionX 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 ) ≤ 2 and S is open, and we show that this result becomes false in general when dim(V ) ≥ 3.
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