AI 中文总结
本文在混合塔中移除封闭性假设，将并行Risch方法推广至任意位置的简单根式扩张，给出统一的赋值引理与次数界，并返回非初等性的证书。
AI 中文摘要
在第一部分中，我们将支撑Risch--Norman（并行Risch）方法的结构定理推广到微分域$K=F(t_1,\dots,t_n)$（在推导下封闭）的简单根式扩张$L=K(y)$，$y^m=q$。在此，我们移除封闭性假设：根式可以位于塔中的任意位置，因此其上方生成元的导数涉及$y$——即Bronstein混合初等函数算法的设定。工作环为$\cA=\cO[t_{j+1},\dots,t_n]$，即$F[t_1,\dots,t_n]$在$L$中的整闭包：它是一个Krull域，在Trager基上作为多项式环的自由模，因此所有因式分解仍发生在唯一分解整环中。推导的分母不再是元素，而是$\cA$上的一个因子$\fd_D$，赋值引理在正规高度一素理想处取统一形式$v_P(Dg)=v_P(g)-(1+v_P(\fd_D))$，涵盖了第一部分中的位移$\{1,e_P\}$；其证明是局部化的，无需消去分析。类群和单位群的稳定性，$\Cl(\cA)\cong\Cl(\cO)$及$\cA^*=\cO^*$，将可容许的对数项分为$\cO$的$S$-单位（由第一部分的机制计算）和在上层变量中移动的不可约多项式，其留数必须是常数。我们证明了顶层变量的次数界，并描述了所得算法——与经典并行方法（其失败不证明任何事）不同，该算法在两种情况下返回非初等性的证书：留数在常数域之外，以及当所有生效的界均被证明时，线性系统显示无留数余项非恰当。
英文摘要
In Part I we extended the structure theorems underlying the Risch--Norman (parallel Risch) method to a simple radical extension $L=K(y)$, $y^m=q$, of a differential field $K=F(t_1,\dots,t_n)$ closed under the derivation. Here we remove the closure hypothesis: the radical may occupy any position in the tower, so that the derivatives of the generators above it involve $y$ --- the setting of Bronstein's algorithm for mixed elementary functions. The working ring is $\cA=\cO[t_{j+1},\dots,t_n]$, the integral closure of $F[t_1,\dots,t_n]$ in $L$: a Krull domain, free over the polynomial ring on Trager's basis, so that all factorisation remains in a unique factorisation domain. The denominator of the derivation is no longer an element but a divisor $\fd_D$ on $\cA$, and the valuation lemma takes the unified form $v_P(Dg)=v_P(g)-(1+v_P(\fd_D))$ at normal height-one primes, subsuming the shifts $\{1,e_P\}$ of Part I; the proof localises and requires no cancellation analysis. Stability of the class group and the unit group, $\Cl(\cA)\cong\Cl(\cO)$ and $\cA^*=\cO^*$, splits the admissible logands into $S$-units of $\cO$ --- computed by the machinery of Part I --- and irreducible polynomials moving in the upper variables, whose residues must be constants. We prove degree bounds in the top variable and describe the resulting algorithm, which --- unlike the classical parallel method, whose failure proves nothing --- returns certificates of non-elementarity in two situations: a residue outside the constant field, and, when every bound in force is proved, a residue-free remainder that the linear system shows to be non-exact.
