AI 中文总结
该研究将并行Risch方法的积分结构事实扩展到简单根式扩张，明确了积分分母形式与可容许对数项，关联了S-单位与雅可比挠元及多项式Pell方程，给出了亏格0对数部分的完整描述并构建了算法。
AI 中文摘要
并行Risch（Risch-Norman）方法是一种用于计算超越扩张塔上初等积分的快速启发式算法，其依据是积分的两个结构事实：分母的界和可能出现的对数的描述。对于纯对数塔，这两个事实已由Davenport-Trager证明；对于多元有理函数域上的任意导子，其结构定理形式的相关结论由Bronstein给出。我们将这两个事实扩展到此类域的简单根式扩张L=K(y)（满足y^m=q）的情形。关键观察在于：F[t₁,…,tₙ]在L中的整闭包具有显式基，因此所有因式分解均可在多项式环中进行；且导子在每个高度1素理想P处有良定义的极点阶δ_P∈{0,1,e_P}，导数的极点阶会按δ_P偏移。积分的分母具有与超越情形相同的Hermite型形式，而可容许的对数项恰好是整闭包的S-单位，其中S为显式有限素理想集合；后者可能大于不可约多项式生成的集合，例如单位x+√(x²+1)所示。对于n=1，我们将这些S-单位与雅可比簇中的挠元关联；对于m=2，将其与多项式Pell方程关联，从而得到亏格0情形下对数部分的完整描述。我们描述了由此得到的算法并给出了示例。
英文摘要
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.
