User sonat suer - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:57:04Z http://mathoverflow.net/feeds/user/2513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8579/are-all-polynomial-inequalities-deducible-from-the-trivial-inequality/8595#8595 Answer by Sonat Suer for Are all polynomial inequalities deducible from the trivial inequality? Sonat Suer 2009-12-11T19:43:42Z 2009-12-11T19:43:42Z <p>Considering the word <em>deduce</em>, I think a possible way to address this question is to use logic, more precisely model theory. Here is a model theoretic restatement of your question: What is a good axiomatization of the quantifier free theory of reals as an ordered ring? The answer is a classical theorem of Tarski, which also solves Hilbert's 17th problem with some algebra. Tarski's theorem says that the theory of real closed fields has quantifier elimination. This implies that the theorey of real closed fields is complete as there is an algebraicaly prime model, namely the rationals. So a valid system of polynomial inequalities in any real closed field is deducible from the axioms of real cosed fields.</p>