Ax-Schanuel for Linear Differential Equations

TL;DR

This paper extends the Ax-Schanuel theorem to linear differential equations with constant coefficients, providing a comprehensive axiomatization of their first-order theories and demonstrating the effectiveness of generalized inequalities.

Contribution

It generalizes the exponential Ax-Schanuel theorem to all linear differential equations with constant coefficients and offers a complete axiomatization of their first-order theories.

Findings

Generalization of Ax-Schanuel inequalities to linear differential equations

Complete axiomatization of first-order theories for these equations

Validation of inequalities as adequate for the theories

Abstract

We generalise the exponential Ax-Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by J. Kirby and C. Crampin we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax-Schanuel inequalities are adequate for them.