The ring of bounded polynomials on a semi-algebraic set

Abstract

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 B V (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 completion X of V yields a ring isomorphism O U (U ) ∼ → B V (S) for some (concretely specified) open subvariety U ⊃ V of X. We prove that B V (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.

Toric completions and bounded functions on real algebraic varieties

“…In dimensions ≥ 3, it is not even guaranteed that the ring B(S) is finitely generated (see [11] Sect. 5).…”

“…Compatible completions were introduced by the authors in [11], as well as in the dissertation of the first author, motivated by earlier work of Powers and Scheiderer in [12]. An S-compatible completion of V yields in particular a description of the ring of regular functions on V that are bounded on S. If V ֒→ X is such an S-compatible completion and Y is the union of those irreducible components of X V that are disjoint from S, then B V (S) is naturally identified with O X (X Y ), the ring of regular functions of the variety X Y .…”

“…The main goal of this paper is to improve on the results in [11] and make them more explicit in the controlled setting of toric varieties. Specifically, we study the following questions.…”

“…(1) One of the main results of [11] is the existence of regular completions in the case dim V ≤ 2. The higher-dimensional case remains open and hinges on the existence of a certain type of embedded resolution of singularities.…”

“…However, it does not imply that B V (S) is a finitely generated R-algebra. This was discussed in [11] and much further explored by Krug in [5]. When a toric S-compatible completion exists, B V (S) is always finitely generated (Proposition 2.10).…”