A Ganzstellensatz for Semialgebraic Sets and a Boundedness Criterion for Rational Functions

Abstract

Let ⟨ K, ν ⟩ be a real closed valued field, and let S ⊆ K ⁿ be an open semialgebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on S, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semialgebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure.

Key Words:

• Algebra
• Model theory
• Real algebraic geometry
• Valuation theory

2010 Mathematics Subject Classification:

• Primary 14P10
• Secondary 03C64
• 12J20

ACKNOWLEDGMENTS

The main part of this work was submitted as part of the author's M.Sc. thesis, supervised by Assaf Hasson at Ben-Gurion University, Israel. The author wishes to thank, from the bottom of her heart Yoav Yaffe for introducing her the question of this work and the Baer–Krull ingredient which has been central in this work, and for his devoted guidance without which the work probably would not have taken place. The main body of this article (Sections 2–5) represents joint work with (and was jointly written by) Yoav Yaffe, who supervised informally on the master thesis, and out of his own reasons and time constraints chose not to be a co-author of this article. The author would like also to thank Jeff Burdges, Antongiulio Fornasiero, and Liran Shaul for carefully reading the article, and giving their useful comments.

Notes

Communicated by A. Prestel.