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

Let ⟨ K, ν ⟩ be a real closed valued field, and letS ⊆ Knbe an open semialgebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, onS, 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. [ABSTRACT FROM AUTHOR]