Parallel Integration over Simple Radical Extensions

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 $\delta_P\in\{0,1,e_P\}$ at every height-one prime $P$, so that pole orders of derivatives shift by $\delta_P$. 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.