On preordered fields related Hilbert's 17th problem
This paper is a continuation of the author's earlier paper [Proc. Am. Math. Soc. 104, 335-342 (1988)]. In this paper, we continue to investigate preordered fields with the Weak Hilbert Property, for which the version of Hilbert's 17th problem on preordered fields is affirmative. By introducing another new notion ``strongly local density'', we show that a preordered field has the Weak Hilbert Property if and only if it is strongly locally dense. As a succedent result, we give again a characterization of preordered fields with the Weak Hilbert Property via special real places, which just generalizes the crucial implication in the definition of Weak Hilbert Property from rational to arbitrary function fields. Moreover, by the preceding result, we show the following result: Let both (K,S) and \((K^*,S^*)\) be preordered fields, and let \((K^*,S^*)\) be a finitely generated extension of (K,S). If (K,S) has the Weak Hilbert Property, then \((K^*,S^*)\) has also the Weak Hilbert Property. Finally, we point out the fact that the family of preordered fields with the Weak Hilbert Property is not inductive.
scientific article; zbMATH DE number 4130498