Orderings and real places on commutative rings

Abstract

Over fields, there is a close relationship between orderings and real places which has been exploited in quadratic form theory [3,5,6, 131 and in the study of the real holomorphy rings and sums of 2n-powers [1,2, 21,221. Also, orderings on an arbitrary commutative ring [4, 141 have proved useful in real algebraic geometry and in quadratic form theory over rings more general than fields. Of course, real places and valuations on fields have played an important role in this theory too. At the same time, there is a general theory of valuations on commutative rings which has been in existence for some time [9, 10,15,23,24]. In this paper, k-places, k a formally real field (in particular, R-places) are considered on an arbitrary commutative ring R. To obtain the good separation properties we want, the places we consider are functions