Elementary Integration of Superelliptic Integrals

TL;DR

This paper introduces an algorithm for elementary integration of superelliptic integrals, capable of solving a broad class of such integrals efficiently under generic conditions.

Contribution

It presents a novel algorithm that solves the elementary integration problem for superelliptic integrals with specific algebraic conditions, improving computational efficiency.

Findings

Algorithm solves the integration problem in polynomial time for generic cases.

Complexity depends on degree, genus, and coefficient height of the polynomial.

Provides a theoretical foundation for symbolic integration of superelliptic functions.

Abstract

Consider a superelliptic integral $I=\int P/(Q S^{1/k}) dx$ with $\mathbb{K}=\mathbb{Q}(\xi)$, $\xi$ a primitive $k$th root of unity, $P,Q,S\in\mathbb{K}[x]$ 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 $O((kd)^{\omega+2g+1} h^{g+1})$ operations.