The theory of the exponential differential equations of semiabelian varieties

TL;DR

This paper characterizes the complete first order theories of exponential differential equations on semiabelian varieties, linking model theory, algebra, and differential equations, and generalizing Ax's conjecture.

Contribution

It provides a new model-theoretic framework for exponential differential equations on semiabelian varieties, including an amalgamation construction and a generalized Schanuel's conjecture.

Findings

Complete first order theories of these equations are described.

Necessary and sufficient conditions for solutions are established.

A purely algebraic corollary, the 'Weak CIT', is derived.

Abstract

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.