In search of a complete axiomatization for $F_p((t))$
University of Katowice
I will give a survey of the attempts that have been made since the mid 1960’s to find a complete recursive axiomatization of the elementary theory of $F_p((t))$. This problem is still open, and I will describe the difficulties researchers have met in their search. Some new hope has been generated by Yu. Ershov’s observation that $F_p((t))$ is an “extremal” valued field. However, while his intuition was good, his definition of this notion was flawed. It has been corrected in a paper by Azgin, Kuhlmann and Pop, in which also a partial characterization of extremal fields was given. Further progress has been made in a recent manuscript, on which I will report at the AMS meeting at Rutgers. The talk at the Graduate Center will provide a detailed background from the model theoretic point of view.
The property of being an “extremal valued field” is both elementary and very natural, so it is an ideal candidate for inclusion in a (hopefully) complete recursive axiomatization for $F_p((t))$. It implies an axiom scheme that was considered previously, which describes the behavior of additive polynomials under the valuation. I will discuss why additive polynomials are crucial for the model theory of valued fields of positive characteristic.
The open problems around extremal fields provide a good source of research projects of various levels of difficulty for young researchers.
This talk is jointly sponsored by the Commutative Algebra & Algebraic Geometry Seminar and the CUNY Logic Workshop.