| Résume | Joint work of Matthias Aschenbrenner, Joris van der Hoeven, and me led to the following two theorems about maximal Hardy fields:
(1) they are all elementarily equivalent to the ordered differential field of transseries;
(2) they are η_1 in the sense of Hausdorff.
This happened several years ago. As to (1), the proof goes through with “Hardy field” replaced by “analytic Hardy field” (with corresponding notion of “maximal”). This was not the case for (2), where we used gluing constructions and partitions of unity unavailable in the analytic context. Last year, Aschenbrenner and I did establish (2) also in the analytic case by reduction to the non-analytic setting, using Whitney's powerful approximation theorem. I will give an overview of this, recalling also the background about transseries and asymptotic differential algebra. There are further things to say about analytic Hardy fields that have no obvious analogue for arbitrary Hardy fields, such as analytic continuation to the complex plane. The second part of my talk will be about that. Some of this, in particular possible connections to o-minimality, will be partly speculative. |
