Real closed fields with non-standard and standard analytic structure

Abstract

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 tadically convergent power series, thus combining the analytic structures from [DD] and [LR3]. 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 [HP] of a sentence which is not true in any o-minimal expansion of R (shown in [LR3] 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, R 1 , . . . , R n−1 .

Fields with analytic structure

“…Since B n,α ⊂ B n,α for each n, α, it is enough to prove that K is o-minimal in L B . Now one can analyze definable functions in one variable (using annuli and the fact that B is strong) exactly as in [CLR2,Section 3], yielding o-minimality as in [CLR2]. The strongness assumption (Definition 3.1.2) is used implicitly in [CLR2,Section 3].…”

“…To prove o-minimality for real closed fields with (not necessarily strong) analytic structure it is sufficient to prove o-minimality for real closed fields with strong analytic structure. The proof of o-minimality for real closed fields with strong analytic structure is given in [CLR2,Section 3], where condition (c) is used (implicitly), roughly speaking to write f (x, 1/x) = g(x) + (1/x)h(1/x).…”

“…Then we are in the context of Section 2 of [CLR2] where we considered the completion of the field of Puiseux series with real analytic B-structure. If we take…”

Real closed valued fields with analytic structure

“…Since the pioneering work of Denef and van den Dries in [6], subanalytic sets and fields with analytic structure have been intensively studied by various authors (see [2,4,5,7,9,11,12,[14][15][16]). The most complete account to date is the one given by Cluckers and Lipshitz in [3], where, inspired by almost all previous work, they provided an abstract setting which isolates the properties that a ring of functions should satisfy in order to behave like a ring of analytic functions.…”

“…In previous work on real closed fields with analytic structure, the analytic structure is in some way intrinsic to the ordered structure. One main thrust of research, which is in the same spirit as this investigation, is that of the structures studied by Lipshitz and Robinson in [9] and later generalized by Cluckers et al in [5] and Cluckers and Lipshitz in [3]. In this setting, real closed valued fields are studied in a language without the valuation, and the c 2019 The Edinburgh Mathematical Society.…”