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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The ring of bounded polynomials on a semi-algebraic set


Authors: Daniel Plaumann and Claus Scheiderer
Journal: Trans. Amer. Math. Soc. 364 (2012), 4663-4682
MSC (2010): Primary 14P99; Secondary 14C20, 14E15, 14P05
Published electronically: April 17, 2012
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Abstract: Let $ V$ be a normal affine $ \mathbb{R}$-variety, and let $ S$ be a semi-algebraic subset of $ V(\mathbb{R})$ which is Zariski dense in $ V$. We study the subring $ B_V (S)$ of $ \mathbb{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)\le 2$ or $ S=V(\mathbb{R})$. An $ S$-compatible completion $ X$ of $ V$ yields a ring isomorphism $ \mathscr {O}_U(U)\overset {\sim }{\to } B_V(S)$ for some (concretely specified) open subvariety $ U\supset V$ of $ X$. We prove that $ B_V(S)$ is a finitely generated $ \mathbb{R}$-algebra if $ \dim (V)\le 2$ and $ S$ is open, and we show that this result becomes false in general when $ \dim (V)\ge 3$.


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Additional Information

Daniel Plaumann
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
Email: daniel.plaumann@uni-konstanz.de

Claus Scheiderer
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
Email: claus.scheiderer@uni-konstanz.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05443-2
PII: S 0002-9947(2012)05443-2
Keywords: Real algebraic varieties, bounded polynomials, normal varieties, divisors, finite generation of rings of global sections, algebraic surfaces
Received by editor(s): February 9, 2010
Received by editor(s) in revised form: July 29, 2010
Published electronically: April 17, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.