The ring of bounded polynomials on a semi-algebraic set

TL;DR

This paper investigates the structure of bounded polynomial rings on semi-algebraic sets within real algebraic varieties, introducing S-compatible completions and analyzing their properties and finite generation in low dimensions.

Contribution

It introduces the concept of S-compatible completions for real algebraic varieties and establishes finite generation of bounded polynomial rings in low dimensions, with counterexamples in higher dimensions.

Findings

Existence of S-compatible completions for varieties of dimension ≤ 2

Isomorphism between B(S) and regular functions on open subvarieties of completions

Finite generation of B(S) in low dimensions, failure in higher dimensions

Abstract

Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of 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 V is of dimension at most 2 or S=V(R). An S-compatible completion X of V yields an isomorphism of B(S) with the ring of regular functions on some (concretely specified) open subvariety of X. We prove that B(S) is a finitely generated R-algebra if S is open and of dimension at most 2, and we show that this result becomes false in higher dimensions.