Improved method for integrals with mixed radical extensions in algebraic towers
Parallel Integration over Simple Radical Extensions II: Mixed Towers
Symbolic Computation
Summary
Calculating certain complicated integrals often involves working with nested functions built from roots and other operations. The authors extended previous work to handle cases where the root functions can appear anywhere within these nested towers, a more general and challenging situation. They created a framework that guarantees unique factorization for the algebraic objects involved, which helps properly analyze these integrals. Their method can also prove when no simpler, 'elementary' integral exists, giving a clear certificate instead of just failing silently.
What this means in practice
- •For symbolic computation developers: Design improved computer algebra system modules to integrate complex functions involving nested roots with guaranteed detection of unsolvable cases.
- •For mathematical software engineers: Implement new algorithms for symbolic integration that provide certificates when integrals cannot be expressed in elementary terms, enhancing reliability.
Authors
Sam Blake
Abstract
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.