Model Theory of Fields With Operators - a Survey

The model theory of fields with operators has proven to be very useful in applications of model theory to problems outside logic. Differentially closed fields and separably closed fields were instrumental in Hrushovski’s proof of the conjecture of Mordell-Lang ([39]), and in work of Hrushovski and Pillay on counting the number of transcendental points on certain varieties ([44]). Let me also mention Ax’s work ([1]) on connections between transcendence theory and differential algebra, which has been recently elaborated on by Bertrand and Pillay ([6]) and Kowalski ([51]) in positive characteristic. The model theory of difference fields also provided various achievements: Hrushovski’s new proof of the conjecture of Manin-Mumford ([40]), Scanlon’s approach to p-adic abc conjectures and proximity questions ([89], [90], [91]), and his solution to Denis’ conjecture ([92]); more recently, applications to algebraic dynamics by Hrushovski and the author ([21], [22], [16]), and by Medvedev and Scanlon ([61]). Model theory also provides some insight on the Galois theory of systems of differential (or difference) equations and of Picard-Vessiot extensions, or of strongly normal extensions of Kolchin, see e.g. [74]. The aim of this article is not to present these applications, but to give a survey of what is known of the model theory of these enriched fields. We will discuss such issues as existence of model companions and their various axiomatisations, elimination of quantifiers and decidability; we will also investigate stability theoretic properties and mention some open problems. Another notable omission of this survey is that of exponential fields, which have seen many extraordinary developments in the past twenty years: in the context of the field of real numbers, or in the context of the field of complex numbers. I do not consider myself an expert in this subject, parts of which are still in full progress.