Symbolic integration on planar differential foliations

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.