The intersection theorem for orderings of higher level in rings

Abstract

This short note is ment as a supplement to the paper “On Rings admitting Orderings and 2-primary Orderings of Higher Level” by E. Becker and D. Gondard ([4]), where an intersection theorem for 2-primary orderings of higher level has been proved ([4]), Proposition 2.6). We will show that the same characterization holds for orderings of arbitrary level. This result finds several applications. For example, it is useful for the continuous representation of sums of 2n-th powers in function fields (see [8]) and it can be applied to derive several Null- and Positivstellensätze for generalized real closed fields (see [5]). As a further example we will prove a strict “Positivstellensatz of higher level” for a certain class of formally real fields. For unexplained notions we refer the reader to [4].