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Model Theory of Differential Fields

Published online by Cambridge University Press:  26 June 2025

Deirdre Haskell
Affiliation:
College of the Holy Cross, Massachusetts
Anand Pillay
Affiliation:
University of Illinois, Urbana-Champaign
Charles Steinhorn
Affiliation:
Vassar College, New York
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Summary

This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications.

A differential field is a field K equipped with a derivation δ : K K; recall that this means that, for x,y K, we have δ (x + y) = δ (x) + 6(y) and δ (xy) = (y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use model-theoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information.

In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski's proof of the Mordell-Lang conjecture in characteristic zero — showing how model-theoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1-3 can be found in [Marker et al. 1996], while the material in Section 4 can be found in [Hrushovski and Sokolovic ≥ 2000; Pillay 1996]. The primary reference on differential algebra is [Kolchin 1973], though the very readable [Kaplansky 1957] contains most of the basics needed here, as does the more recent [Magid 1994]. The book [Buium 1994] also contains an introduction to differential algebra and its connections to diophantine geometry. We refer the reader to these sources for references to the original literature.

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Publisher: Cambridge University Press
Print publication year: 2000

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