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Parallel Integration over Simple Radical Extensions

Sam Blake

arXiv:2608.29482v2cs.SC

TL;DR

The paper addresses whether the structural guarantees behind the heuristic parallel Risch–Norman method extend from rational-function fields to simple radical extensions. It constructs an explicit integral basis and analyzes valuation shifts, showing that denominator bounds persist while logarithms become S-units; in one variable, these are further characterized through Jacobian torsion and polynomial Pell equations. The resulting degree results make the method complete in the stated univariate setting, while tower lower variables retain explicit obstructions.

  • Problem

    The parallel Risch–Norman method lacks general structural guarantees for elementary integrals over simple radical extensions, especially for denominators, logarithms, and numerator degree bounds.

  • Method

    The paper studies the integral closure and height-one valuations of L = K(y), y^m = q, retaining factorisation in R and characterizing logarithms through S-units.

  • Results

    The Hermite-type denominator structure extends to radical extensions; for n = 1, exact numerator degree bounds make the method complete over F̄(x, y), while tower bounds transfer only in the top variable.

  • Takeaways & Limitations

    For n = 1, the logarithmic part reduces to Jacobian torsion, polynomial Pell equations when m = 2, and a complete genus-0 description.

  • Takeaways & Limitations

    The finite S-unit generating set depends on the integrand denominator’s support, and the classical “+1” degree bound can fail for lower variables in towers.

Abstract

from arXiv · show

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.

1 Introduction

The paper extends the structural basis of the parallel Risch–Norman method from rational-function fields to simple radical extensions. It preserves polynomial factorisation and Hermite-type denominator bounds while replacing polynomial logands with S-units and establishing degree results for univariate and tower settings.

  • Motivation: The Risch–Norman method simultaneously guesses integral denominators and logarithm arguments, bounds the polynomial numerator, and solves a linear system, but remains heuristic.It is used as a preprocessor for, or instead of, the complete Risch algorithm in computer algebra systems.
  • Contribution: The paper shows that the method’s two structural foundations extend to L = K(y), y^m = q, while identifying the changes caused by algebraic extension.The extension concerns both denominator bounds and the description of possible logarithms.
  • Integral basis: The integral closure has an explicit basis, giving every element a canonical denominator while keeping gcd, squarefree, and splitting factorisations inside the UFD R.This extends Trager’s univariate basis argument to the multivariate setting.
  • Denominator bounds: At each height-one prime, the derivation has pole order δ_P in {0, 1, e_P}, yielding the same Hermite-type denominator shape as in the transcendental case.The valuation shift replaces key results previously established for logarithmic towers and rational-function fields.
  • Logarithms: The admissible logarithms are S-units of the integral closure rather than merely irreducible polynomials, because the closure may be non-UFD and contain nonconstant units.For example, x + sqrt(x^2 + 1) is a norm-one unit.
  • Degree bounds: For n = 1, exact numerator degree bounds make the method complete over F̄(x, y), while tower results transfer bounds in the top variable but expose failures for lower variables.A lower-variable integral can have degree two more than the integrand, violating the classical “+1” bound.

2 Preliminaries

The preliminaries establish the differential-field setting, the polynomial UFD used for factorisation, and the strong Liouville representation underlying elementary integration. They also define normal and special irreducible factors through the derivation.

  • Differential fields: The paper assumes characteristic-zero fields and works with K = F(t_1, ..., t_n), constants F, and R = F[t_1, ..., t_n].Each t_i is transcendental over the field generated by earlier variables, and R is a UFD.
  • Derivation: The derivation denominator is the least common multiple of the denominators of Dt_i, and multiplying D by it produces a derivation of R.This permits polynomial factorisation to be used in the differential setting.
  • Prime classification: An irreducible p is normal when gcd(p, D_Rp) = 1 and special when p divides D_Rp.Every irreducible is classified into one of these two cases, supporting splitting factorisation.
  • Elementary integration: The strong Liouville theorem represents an elementary integral through a differential-field element, constant coefficients, and logarithmic terms.The supplied preliminaries state this representation before developing the radical extension.

3 Simple radical extensions and their integral basis

The paper constructs the integral closure for a simple radical extension using an explicit polynomial basis and derives its local ramification and valuation structure. These results provide canonical denominators and preserve factorisation in the base UFD.

  • Normalisations: The radical extension is L = K(y) with y^m = q, where q has a squarefree decomposition and is normalized to be m-th-power-free.A second normalization requires y^m − q to remain irreducible over the algebraic closure of the constant field.
  • Normalisations: Under irreducibility normalization, the constant field remains algebraically closed in L and Const_D(L) = F.The proof uses the absence of smaller radical subextensions and the algebraicity of constants in algebraic extensions.
  • Integral basis: The integral closure O is a free R-module with an explicit basis of integral radical monomials whose exponents are determined by the squarefree decomposition of q.The basis is shown to be closed under multiplication and locally integrally closed in codimension one.
  • Denominators: The explicit basis is a local integral basis, so O is the integral closure and every element of L has a canonical denominator in R.The denominator representation uses coefficients in R with a normalized common denominator.
  • Local structure: At an unramified prime p, e_P = 1 and valuations restrict from R; at a branch prime dividing Q_j, e_P = m/gcd(j,m) and the basis valuations are integral.These formulas describe how base valuations and radical valuations combine over height-one primes.
  • Derivation: The squarefree part of q contributes to the denominator of the derivation, paralleling the logarithmic case.This incorporates radical branching into the differential denominator structure.

4 Valuations and the derivation

The section classifies derivation behavior at height-one primes through a pole order δP, yielding precise valuation shifts for derivatives. Normal unramified, normal branch, and special primes form distinct regimes that support the later denominator and logarithm analysis.

  • Valuation shift: For every g∈L*, vP(Dg) is at least vP(g)−δP, with equality whenever vP(g)≠0.Equivalently, Dg/g has leading term vP(g)Dπ/π up to a remainder of valuation at least 1−δP.
  • Unramified primes: The unramified case uses π=p and gives δP=1 because Dp/p has valuation −1.Integral-basis elements and their derivatives remain P-integral in this case.
  • Branch primes: At a branch prime p|Q, a uniformiser can be constructed from y and p, and the derivative has pole order eP.The leading term has valuation −eP, while the remaining contribution is P-integral.
  • Special and non-tame primes: If p is special in R, then Dp/p and Dπ/π are P-integral, so derivatives do not acquire higher-order poles and cancellation remains possible.Primes over factors of denD(K) are non-tame and are absorbed into a guessed special denominator.
  • Pole-order regimes: At every tame height-one prime P, δP is 1 for unramified primes and eP for branch primes, while special primes have δP=0.The three regimes are unified by interpreting δP as the pole order of the logarithmic derivative.

5 Structure of elementary integrals

The section gives the structural decomposition needed by parallel integration over a radical extension: rational denominators obey valuation-based Hermite bounds, while logarithmic terms are represented by S-units of the integral closure. These results preserve polynomial-ring factorisation but make the admissible logarithms depend on the integrand’s denominator support.

  • Denominator structure: At a tame prime P, the rational-part denominator loses one valuation step: vP(v)=vP(f)+δP when vP(f)<−δP, and vP(v)≥0 otherwise.If vP(f)>−δP, every logarithmic unit has valuation zero at P.
  • Denominator structure: The rational part has a Hermite-type denominator in which normal factors are reduced by one multiplicity step, while special factors remain unconstrained.Branch primes are included among the normal factors when their underlying polynomial is normal.
  • Logarithmic structure: Every logarithmic term can be chosen as an Sf-unit of the integral closure, with valuations zero outside the finite prime set Sf.For tame primes, the relevant valuations are determined by the coefficient of the π−δP term after any required local derivative reduction.
  • Logarithmic structure: When the integrand has a sufficiently mild pole at P, the logarithmic valuations are read directly from the π−δP coefficient; larger poles require subtracting a local derivative first.For n=1, the resulting invariant is the ordinary residue of f dx.
  • New algebraic phenomena: Unlike the UFD transcendental case, radical extensions may require separating primes over one polynomial and using non-polynomial units as logands.The example involving √(x−1) shows that an irreducible factor alone may not provide the needed divisor pattern.
  • Scope of the logand set: A finite generating set of logands exists for each finite Sf, but it depends on the support of the integrand denominator rather than only on the field.Thus Davenport–Trager’s field-only formulation of the finite logand set does not persist for radical extensions.

6 The case n = 1: residues, S-units and the Jacobian

For n=1, the paper characterizes elementary integrability through residues, torsion divisor classes in the Jacobian, and S-units of the integral closure. In genus 0 this determines the logarithmic part completely, while m=2 connects the required units to polynomial Pell equations and continued fractions.

  • The affine places are height-one primes of the integral closure, while the places over infinity number gcd(m, deg q), each with ramification index m/gcd(m, deg q).
  • S-units and the Jacobian: For an elementary integral, each nonzero residue value determines a degree-zero divisor class [D_c] that must be torsion in Jac(X).
  • S-units and the Jacobian: The logarithmic factors arise from elements u_c whose divisors are multiples of the residue divisors, while the remaining differential must be the derivative of a rational part.
  • S-units and the Jacobian: Condition (ii) is independent of torsion: a residue-compatible logarithmic derivative can leave a holomorphic discrepancy, so torsion alone does not guarantee elementary integrability.
  • Genus 0 and Pell equations: In genus 0 every relevant divisor is principal, so residues completely determine the logarithmic part and the required units follow from Riemann–Roch linear algebra.
  • Genus 0 and Pell equations: For m=2, nontrivial S-units correspond to polynomial Pell solutions a^2−qb^2=1, equivalently periodic continued fractions of √q when deg q is even.

7 Degree bounds

The degree-bound analysis converts valuation behavior of the derivation into exact numerator bounds for n=1 and a transfer theorem for the top variable of towers. Lower variables remain problematic because residue-derivation cancellation can invalidate the classical “+1” bound.

  • The unresolved Risch–Norman condition (ii) is a bound on the rational numerator degree; the paper proves exact n=1 bounds, transfers top-variable bounds, and exhibits lower-variable obstructions.
  • The case n = 1: At every place, derivative valuations satisfy v_P(Dg)=v_P(g)−1+v_P(Dπ) when v_P(g)≠0, with affine and infinite-place shifts determined by the derivation’s pole order.
  • The case n = 1: For n=1, valuation bounds at infinity yield exact numerator-degree bounds and make the parallel ansatz complete when the required torsion orders are bounded.
  • Towers: the top variable: For towers, the top-variable bound transfers under primitive or hyperexponential extensions, preserving the corresponding +1 or 0 shift under the stated hypotheses.
  • Towers: the top variable: The transfer theorem requires an algebraically closed constant field, constant leading coefficient q_N, and a primitive or hyperexponential top variable.
  • Lower variables: the obstruction: For lower logarithmic variables, cancellation may occur when the derivative of the variable lies in the image of the residue derivation, so the classical “+1” bound can fail.
  • Lower variables: the obstruction: The examples show that the radical extension is not itself the obstruction; controlling the residue derivation and its enlarged constant field is the remaining issue.

8 The algorithm

The algorithm combines the explicit radical integral basis, valuation-based denominator and degree bounds, and several tiers for finding S-units. It is exact for n=1 under the torsion conditions, while tower implementations retain heuristic lower-variable bounds and may face special-denominator failures.

  • Polynomial factorization, special-prime selection, and denominator setup remain in the base polynomial ring, while n=1 uses exact bounds and towers use top-variable transfer bounds.
  • Degree bounds: For towers, lower-variable numerator bounds retain Bronstein’s heuristic coordinatewise form because the classical bound can fail there.
  • The integration ansatz differentiates basis terms and logarithms, clears denominators, equates coefficients, and solves a linear system over F.
  • Finding S-units: HiddenUnits offers continued fractions for n=1,m=2, bounded-degree norm-constrained searches in general, and an exact residues–torsion–Riemann–Roch tier for n=1.
  • Caveat: The guessed special denominator is not universally bounded when a branch prime is special or divides the base derivation denominator.

9 Examples

The examples demonstrate that exact radical integration requires non-polynomial logands, including units at infinity and units associated with separated places. They also show successful handling of nontrivial integral bases and special exponential radicals.

  • The prototype uses exact arithmetic and residue computations, searches for divisor-producing elements, and obtains m=2 units at infinity from continued fractions.
  • Genus 0: In the genus-0 case y^2=x^2+1, restricting logands to irreducible polynomial factors gives no solution because the required places over x=2 must be separated.
  • Genus 1: For y^2=x^4+1, the periodic continued fraction yields the unit x^2+y of norm −1, and the solver returns a logarithm involving 1+y.
  • Cubic radical: For y^3=x^2, three residue values require three logarithms, and omitting the constant ω from the logand set prevents the linear system from having a solution.
  • Exponential radical: For the exponential radical y=e^(x/2), the setup identifies the branch prime as special and reduces to the corresponding algorithm for Q(x,e^(x/2)).

10 Conclusion and open problems

The paper extends the two structural foundations of the parallel Risch method to simple radical extensions, retaining a polynomial factorization setting while replacing polynomial logarithms with S-units of the integral closure. For one variable, the degree bound is exact and the method is complete, but several multiradical, higher-dimensional, and algorithmic questions remain open.

  • Conclusion: The Hermite-type denominator structure and characterization of possible logarithms extend to simple radical extensions of differential fields.Factorization remains in the polynomial ring, while admissible logarithms are S-units of the integral closure rather than irreducible polynomials.
  • Conclusion: For n = 1, the S-unit problem becomes the classical torsion problem in the Jacobian.
  • Open problems: For n = 1, the degree bound is exact and the resulting method is complete.
  • Open problems: Open problems include degree bounds in lower variables, several radicals without a general explicit integral basis, and S-unit generators for n > 1.The lower-variable bound must depend on the residue derivation at t_i = ∞, and the Davenport–Trager form of one condition is false as stated.
  • Open problems: It remains unclear whether the approach can serve as a useful preprocessor for complete algebraic integration algorithms in practice.
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