The ring of bounded polynomials on a semi-algebraic set
Algebraic Geometry
2010-07-30 v2
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.
Cite
@article{arxiv.1002.1870,
title = {The ring of bounded polynomials on a semi-algebraic set},
author = {Daniel Plaumann and Claus Scheiderer},
journal= {arXiv preprint arXiv:1002.1870},
year = {2010}
}