Elementary Integration of Superelliptic Integrals

Abstract

Consider a superelliptic integral I = P/(QS 1/k )dx with K = Q(ξ), ξ a primitive kth root of unity, P, Q, S ∈ K[x] and S has simple roots and degree coprime with k. Note d the maximum of the degree of P, Q, S, h the logarithmic height of the coefficients and g the genus of y k − S(x). We present an algorithm which solves the elementary integration problem of I generically in O((kd) ω+2g+1 h g+1 ) operations.