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arxiv: 2606.30708 · v1 · pith:EU6WYVGFnew · submitted 2026-06-29 · ✦ hep-ph · hep-th· physics.comp-ph

LinApart3: efficient algorithm for multivariate partial fraction decomposition with linear denominators

L. Fek\'esh\'azy , A. Kardos This is my paper

Pith reviewed 2026-07-01 02:12 UTC · model grok-4.3

classification ✦ hep-ph hep-thphysics.comp-ph
keywords partial fraction decompositionmultivariate rational functionslinear denominatorshyperplane arrangementresidue extractionsymbolic computationFeynman integralsparallel algorithms
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The pith

LinApart3 decomposes multivariate rational functions with linear denominators so each term uses at most as many distinct factors as there are partial-fraction variables, introduces no spurious singularities, and yields the same result regar

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an algorithm that takes a rational function whose denominators are all linear and rewrites it as a sum of simpler fractions. The rewritten expression satisfies a strict bound: no summand contains more distinct linear factors than the number of variables being decomposed. The method works by reading off the geometry of the hyperplanes defined by those linear factors and performing the necessary cancellations with linear algebra plus residue extraction instead of Gröbner-basis or ideal-membership calculations. Because the individual contributions are independent, the procedure parallelizes naturally and remains unchanged when extra spectator variables are present.

Core claim

LinApart3 performs multivariate partial fraction decomposition for rational functions with linear denominators by exploiting the hyperplane arrangement geometry, replacing Gröbner basis or Leinartas methods with linear algebra and residue extraction. This yields terms each containing at most as many distinct original denominators as there are partial fraction variables, introduces no spurious singularities, remains independent of variable ordering, and is unaffected by spectator variables. The algorithm is naturally parallelizable due to independent basis contributions.

What carries the argument

The geometry of the hyperplane arrangement defined by the linear denominators, which permits replacement of polynomial-ideal computations by linear algebra plus residue extraction.

If this is right

  • High-dimensional integrals or amplitudes can be reduced symbolically without intermediate expression growth.
  • Parallel implementations become straightforward because each basis contribution can be computed independently.
  • Results remain reproducible across different computer-algebra systems because the output does not depend on variable ordering.
  • Spectator variables can be carried through the decomposition without extra cost or special handling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric replacement strategy might apply to other linear arrangements that appear in multi-loop calculations.
  • Combining the algorithm with existing reduction tools could automate more steps in amplitude computations.
  • Because the method is insensitive to extra variables, it may simplify treatments of multi-scale problems where some parameters are held fixed.
  • Implementation in open-source computer-algebra libraries would allow direct testing on benchmark integrals from the literature.

Load-bearing premise

The geometry of the hyperplane arrangement permits replacement of polynomial-ideal computations by linear algebra plus residue extraction without loss of correctness or introduction of intermediate swell.

What would settle it

A concrete multivariate rational function with linear denominators on which the algorithm produces at least one term containing more distinct denominators than the number of partial-fraction variables or adds a pole absent from the original expression.

Figures

Figures reproduced from arXiv: 2606.30708 by A. Kardos, L. Fek\'esh\'azy.

Figure 1
Figure 1. Figure 1: Timings and memory usage of the new LinApart function, our own implementation of the Leinartas method, the Gröbner basis method and MultivariateApart (denoted as LA, L, G and MA in the legend) in case of differ￾ent rational functions with numeric polynomial coefficients. In Figure 1a we plotted the benchmarks with increasing number of denominators with two variables (n = 2), while in Figure 1b we show the … view at source ↗
Figure 2
Figure 2. Figure 2: Timings and memory usage of the new LinApart function (a) and our own implementation of the Leinartas method (b) in case of different rational functions with numeric polynomial coefficients, two partial fraction variables and different number of spectator variables. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as Figure 2 but with the Gröbner-basis method. [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Timings and memory usage of the new LinApart function, our own implementation of the Leinartas method, the Gröbner basis method and MultivariateApart (denoted as LA, L, G and MA in the legend) in case of differ￾ent rational functions with numeric polynomial coefficients. In Figure 4a and Figure 4b we plotted the benchmarks with increasing number of partial fraction variables with one and three linearly ind… view at source ↗
Figure 5
Figure 5. Figure 5: Timings and memory usage of the new LinApart function, our own implementation of the Leinartas method, the Gröbner basis method and MultivariateApart (denoted as LA, L, G and MA in the legend) in cases of increasing multiplicity of one denominator of a fraction with linearly independent denominators with different variable number and denominator number, where polynomial coefficients were “high” random inte… view at source ↗
Figure 6
Figure 6. Figure 6: Timings and memory usage of the new LinApart function, our own implementation of the Leinartas method, the Gröbner basis method and MultivariateApart (denoted as LA, L, G and MA in the legend) in cases of increas￾ing multiplicity of all denominators of a fraction with linearly independent denom￾inators with different variable number and denominator number, whose polynomial coefficients were “high” random i… view at source ↗
Figure 7
Figure 7. Figure 7: Timings and memory usage of the new LinApart function, our own imple￾mentation of the Gröbner basis method and MultivariateApart (denoted as LA, G and MA in the legend) in cases of increasing order of the numerator with two or three variables and four denominators, with “high” random integer numbers as polynomial coefficients. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Timings and memory usage of the new LinApart function, our own implementation of the Leinartas method, the Gröbner basis method and MultivariateApart (denoted as LA, L, G and MA in the legend) in cases of increasing number of null-relations between the denominators. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the total denominator power [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Per-cell wall time of the cover-first cache branch against the determinant [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Per-P i mi-bin wall time of the determinant branch (red) and the cover-first cache branch (blue), aggregated over the 2743 in-scope dpentagon cells of Ref. [15]. Bars are bin totals on a log scale. The cell count is annotated above each bin’s bar pair. Phase 1 vs Phase 2 breakdown. The wall times reported so far combine the two phases of the algorithm. To see where each branch spends its time, we re-ran t… view at source ↗
Figure 12
Figure 12. Figure 12: Total wall time over the 2743 in-scope dpentagon cells, split into Phase 1 [PITH_FULL_IMAGE:figures/full_fig_p039_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Per-P i mi-bin wall time of the determinant branch (red) and the cover￾first cache branch (blue), split into Phase 1 (left, null-relation elimination) and Phase 2 (right, basis identification and residues), over the 2743 in-scope dpentagon cells. Bars are bin totals on a log scale; the cell count is annotated above each bin’s bar pair. The cache branch wins Phase 1 on the heavy tail and loses Phase 2 thro… view at source ↗
read the original abstract

We present LinApart3, an efficient multivariate partial fraction decomposition algorithm for rational functions with linear denominators. Our decomposition algorithm guarantees that each term contains at most as many distinct denominators from the original set as partial fraction variables, introduces no spurious singularities, is independent of variable ordering, and is insensitive to the presence of spectator variables. While general multivariate approaches based on Gr\"obner bases or Leinartas' method handle arbitrary polynomial denominators, they suffer from intermediate expression swell. LinApart3 replaces polynomial-ideal computations with linear algebra and residue extraction by exploiting the geometry of the hyperplane arrangement defined by the denominators, circumventing this issue just as LinApart did in the univariate case. Because the individual basis contributions are independent, the algorithm is moreover naturally parallelizable. To showcase the utility of our algorithm we implemented the algorithm both in Wolfram Mathematica and FORM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper presents LinApart3, an efficient algorithm for multivariate partial fraction decomposition of rational functions with linear denominators. The algorithm is claimed to guarantee that each term contains at most as many distinct denominators from the original set as there are partial-fraction variables, introduces no spurious singularities, is independent of variable ordering, and is insensitive to spectator variables. It achieves this by replacing Gröbner-basis or Leinartas-style polynomial-ideal computations with linear algebra plus residue extraction, exploiting the geometry of the hyperplane arrangement defined by the denominators. The method is naturally parallelizable, and implementations in both Wolfram Mathematica and FORM are provided to demonstrate utility.

Significance. If the stated guarantees hold and are supported by the full derivation, the work would supply a practical tool for a recurring task in high-energy physics calculations involving multivariate rational functions with linear denominators. Replacing ideal computations by linear algebra avoids the intermediate expression swell that limits general-purpose methods; the parallel structure and dual-language implementations are concrete strengths that aid both performance and reproducibility.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'partial fraction variables' is used without an explicit definition or reference to its meaning in the context of the decomposition; a one-sentence clarification would improve accessibility.
  2. The manuscript would benefit from a short worked example (even a bivariate case) early in the text to illustrate how the hyperplane-geometry step produces the claimed bound on the number of denominators per term.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of LinApart3, the recognition of its practical utility in high-energy physics calculations, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; algorithm is a constructive replacement of ideal methods

full rationale

The paper introduces LinApart3 as a new algorithm that exploits hyperplane geometry to replace Gröbner/ideal computations with linear algebra plus residue extraction. The listed guarantees (at most as many denominators per term as variables, no spurious singularities, ordering independence, spectator insensitivity) are stated as direct consequences of this replacement and the independence of basis contributions. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The univariate LinApart reference is contextual background only and does not justify the multivariate correctness claims. The derivation is therefore self-contained as an explicit algorithmic construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit free parameters, axioms, or invented entities; the method is described as relying on standard linear algebra and residue extraction applied to the geometry of linear hyperplanes.

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Reference graph

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