Integration in finite terms and exponentially algebraic functions

TL;DR

This paper develops new techniques combining differential algebra and model theory to analyze exponential algebraicity and independence of solutions to differential equations, with applications to classical special functions.

Contribution

It introduces generalized methods for exponential transcendence and independence, extending classical techniques to broader systems of differential equations.

Findings

Proves exponential transcendence of certain classical functions.

Establishes independence relations among solutions of differential equations.

Provides decidability results for exponential algebraic relations.

Abstract

We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that is, a holomorphic solution which is definable in the structure of restricted elementary functions? Do solutions of a given list of algebraic differential equations share a nontrivial exponentially algebraic relation, that is, a nontrivial relation definable in the structure of restricted elementary functions? These problems can be traced back to the work of Abel and Liouville on the problem of integration in finite terms. This article concerns generalizations of their techniques adapted to the study of exponential transcendence and independence problems for more general systems of differential equations. As concrete applications, we obtain exponential transcendence and independence statements for several classical functions: the error function, the Bessel functions, indefinite integrals of algebraic expressions involving Lambert's W-function, the equation of the pendulum, as well as corresponding decidability results.