Symbolic integration on planar differential foliations

TL;DR

This paper develops a symbolic integration method for rational functions involving solutions of differential equations, introducing telescopers that generalize elementary integrals and providing algorithms for their computation and applications to Liouvillian solutions.

Contribution

It introduces the concept of telescopers for transcendental solutions of differential equations and provides algorithms to compute them, extending symbolic integration techniques.

Findings

The integral family is either differentially transcendental or satisfies a linear differential equation in the parameter.

An algorithm with specific complexity bounds is provided for computing telescopers.

The method can find Liouvillian solutions of planar rational vector fields when they exist.

Abstract

We consider the problem of symbolic integration of $\int G(x,y(x)) dx$ where $G$ is rational and $y(x)$ is a non algebraic solution of a differential equation $y'(x)=F(x,y(x))$ with $F$ rational. As $y$ is transcendental, the Galois action generates a family of parametrized integrals $I(x,h)=\int G(x,y(x,h)) dx$. We prove that $I(x,h)$ is either differentially transcendental or up to parametrization change satisfies a linear differential equation in $h$ with constant coefficients, called a telescoper. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on their order and degree $\hbox{ord},N$ with complexity $\tilde{O}(N^{\omega+1} \hbox{ord}^{\omega-1}+N\hbox{ord}^{\omega+3})$. For the specific foliation $y=\ln x$, a more complete algorithm without an a priori bound is presented. Oppositely, non existence of telescoper is proven for a classical planar Hamiltonian system. As an application, we present an algorithm which always finds, if they exist, the Liouvillian solutions of a planar rational vector field, given a bound large enough for some notion of complexity height.