Parallel Integration over Simple Radical Extensions II: Mixed Towers

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 $\text{cA}=\text{cO}[t_{j+1},\dots ,t_n]$, 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 ${\text{fd}}_D$ on $\text{cA}$, and the valuation lemma takes the unified form $v_P(Dg)=v_P(g)-(1+v_P({\text{fd}}_D))$ 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, $\text{Cl}(\text{cA})\cong \text{Cl}(\text{cO})$ and ${\text{cA}}^{\ast }={\text{cO}}^{\ast }$, splits the admissible logands into $S$-units of $\text{cO}$ — 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.