Integration in finite terms and exponentially algebraic functions
Rémi Jaoui, Jonathan Kirby
TL;DR
The paper develops a unified framework marrying differential algebra and model theory to address exponential algebraicity questions in the o-minimal setting $\mathbb{R}_{RE}$. It introduces the blurred exponential theory $\mathrm{BE}$ and an associated predimension, exponential hulls, and a pregeometry $\mathrm{ecl}^K$, enabling a Galois-theoretic analysis of exponentially algebraic relations and their solutions. The main results (Theorems A and B) provide a dividing line: for equations internal to constants, exponentially algebraic solutions must be elementary, and the binding groups are linear abelian-by-finite, yielding strong exponential independence statements for classical functions such as $\mathrm{sn}$, $\mathrm{erf}$, $\mathrm{Ai}$, and various Bessel functions, among others. The framework gives decidability consequences for integration in finite terms and delivers concrete applications to the pendulum, Lambert’s equation, and Kepler-type equations, linking Liouville’s integration program with modern model theory and differential Galois theory.
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.