Real closed fields with non‐standard and standard analytic structure

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 ℝ n (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 σ n , true in ℝ n , but not true in any o‐minimal expansion of any of the fields ℝ, ℝ 1 , …, ℝ n −1 .