Integration of multivariate rational functions given by straight-line programs

Abstract

We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of ℚ(X1,…,Xn) as input, decides whether they allow a rationalpotential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.