|Marcus Tressl (University of Regensburg, Germany)|
A Uniform Companion for Large Differential Fields
Abstract: We introduce a first order theory of differential fields of characteristic 0, called UC (for Uniform Companion), with the following properties:
1. Whenever L and M are models of UC and F is a common differential subfield of L and M such that L and M have the same universal theory over F as pure fields, then they have the same universal theory over F as differential fields.
2. Every differential field F which is 'large' can be extended to a model of UC and this extension is elementary in the language of rings. Here, a field F is called large if every smooth curve defined over F that has an F-rational point, has infinitely many F-rational points (e.g. PAC, PRC and PpC-fields are large).
The theory UC is a simultaneous axiomatization of differentially closed fields (introduced by T. McGrail) and of differentially closed ordered fields (introduced by M. Singer in the case of one derivative). If we add UC to the theory of p-adically closed fields in the language of p-valued fields we get the model completion of p-adically closed, differential fields. If we add UC to the theory of pseudo finite fields in the language of rings enlarged by some constants, we get the model companion of pseudo finite, differential fields (of characteristic 0).