A note on the Risch differential equation

Abstract

This paper relates to the technique of integrating a function in a purely transcendental regular elementary Liouville extension by prescribing degree bounds for the transcendentals and then solving linear systems over the constants. The problem of finding such bounds explicitly remains yet to be solved due to the so-called third possibilities in the estimates for the degrees given in R. Risch's original algorithm.

We prove that in the basis case in which we have only exponentials of rational functions, the bounds arising from the third possibilities are again degree bounds of the inputs. This result provides an algorithm for solving the differential equation y′+f′y=g in y where f, g and y are rational functions over an arbitrary constant field. This new algorithm can be regarded as a direct generalization of the algorithm by E. Horowitz for computing the rational part of the integral of a rational function (i.e. f′=0), though its correctness proof is quite different.

This research was partially supported by the National Science and Engineering Council of Canada under grant 3-643-126-90.