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An Isaac Newton Institute Workshop

An Introduction to Recent Applications of Model Theory

Schanuel conditions for Weierstrass differential equations

Author: Jonathan Kirby (Oxford)

Abstract

I will discuss a version of Schanuel's conjecture for Weierstrass equations in differential fields. This gives a necessary and sufficient condition for a system of Weierstrass differential equations to have a solution.

The necessity part builds on work by James Ax, who proved the equivalent statement for the exponential equation, and by Brownawell and Kubota who proved an analogue for complex power series. The sufficiency part builds on work of Cecily Crampin.

I hope also to show connections to the theory of the complex Weierstrass p-functions and to structures constructed via Hrushovski's amalgamation technique.