Parallel Integration over Simple Radical Extensions
Symbolic Computation
Summary
The authors extend important structural results about integrating functions with logarithms to cases involving simple radical extensions, where a new variable satisfies an equation like y^m = q. They show that the integral's denominator and possible logarithm terms have a predictable form, similar to previously known cases but now including radicals. Their approach uses properties of polynomial rings and how derivatives behave at certain special points, leading to a clear description of the integral's logarithmic part. For some specific cases, they connect these results to classic objects in algebraic geometry and number theory, such as the Jacobian and the polynomial Pell equation. Lastly, they provide an algorithm based on these findings along with examples.
Authors
Sam Blake
Abstract
The parallel Risch (Risch--Norman) method is a fast heuristic for computing elementary integrals over towers of transcendental extensions. Its justification rests on two structural facts about the integral: a bound on its denominator and a description of the logarithms that can occur. Both are known for purely logarithmic towers (Davenport--Trager) and, in the form of a structure theorem, for arbitrary derivations on multivariate rational function fields (Bronstein). We extend both facts to a simple radical extension $L=K(y)$, $y^m=q$, of such a field. The key observations are that the integral closure of $F[t_1,\dots,t_n]$ in $L$ has an explicit basis, so that all factorisation can remain in a polynomial ring, and that the derivation has a well-defined pole order $δ_P\in\{0,1,e_P\}$ at every height-one prime $P$, so that pole orders of derivatives shift by $δ_P$. The denominator of the integral then has the same Hermite-type shape as in the transcendental case, while the admissible logands are precisely the $S$-units of the integral closure for an explicit finite set $S$ of primes; the latter can be larger than the set generated by irreducible polynomials, as the unit $x+\sqrt{x^2+1}$ shows. For $n=1$ we relate these $S$-units to torsion in the Jacobian and, for $m=2$, to the polynomial Pell equation, obtaining a complete description of the logarithmic part in genus~0. We describe the resulting algorithm and give examples.