Real closed fields with non-standard and standard analytic structure

Abstract

We consider the ordered field which is the completion of the Puiseux series field over ℝ equipped with a ring of analytic functions on [−1, 1]n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures of Denef and van den Dries [Ann. of Math. 128 (1988) 79–138] and Lipshitz and Robinson [Bull. London Math. Soc. 38 (2006) 897–906]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields ℝn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [J. Symbolic Logic 72 (2007) 119–122] of a sentence which is not true in any o-minimal expansion of ℝ (shown in [Bull. London Math. Soc. 38 (2006) 897–906] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in ℝn, but not true in any o-minimal expansion of any of the fields ℝ, ℝ1, …, ℝn−1.