Rational-function interpolation from p-adic evaluations in scattering amplitude calculations

Herschel A. Chawdhry

REVIEW 3 major objections 4 minor 29 references

Rational-function interpolation from p-adic evaluations in scattering amplitude calculations

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A p-adic one-term-at-a-time interpolation reconstructs scattering-amplitude rational functions directly in partial-fractioned form, using about 25 times fewer probes than finite-field methods and yielding expressions about 134 times…

desk verdict Honest, self-contained-enough proceedings summary of the author's own PRD method; the striking numbers are a single-example demonstration, and the load-bearing candidate-denominator oracle is deferred to [1]. read the letter →

arxiv 2412.19868 v1 pith:ILX2J7JK submitted 2024-12-27 hep-ph hep-thphysics.comp-ph

Herschel A. Chawdhry This is my paper · ORCID

classification hep-phhep-thphysics.comp-ph

keywords partialfractioningp-adicinterpolationrationalfunctionreconstructionscatteringamplitudesmultiloopfinitefieldsnumericalprobesintegration-by-partsreduction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to establish that the large rational functions appearing in multi-loop scattering amplitude calculations can be reconstructed from numerical evaluations directly in partial-fractioned form, one term at a time, using p-adic probes rather than conventional finite-field samples. This matters because partial fractioning is known to shrink these rational functions dramatically, and doing the reconstruction in that form should cut both the number of required probes and the size of the final analytical expression. The paper demonstrates the technique on the largest rational function needed for the full-colour two-loop amplitudes for pp→γγj in massless QCD, reporting about 25 times fewer probes and a final expression roughly two orders of magnitude more compact, while remaining mathematically exactly equal to the original. A sympathetic reader would take away that exploiting the special simplification of loop-amplitude rational functions is feasible and opens a route to cheaper multi-loop calculations.

What carries the argument

The central mechanism is the p-adic isolation of a single denominator. The p-adic absolute value measures divisibility by the prime p, so a polynomial is p-adically 'small' when it is highly divisible by p. If a point x makes one candidate denominator d_k strictly smaller than every other candidate denominator, then the evaluation R(x) equals n_k(x)/d_k(x) plus terms of order 1/$p^{{m-1}}$, as in Eq. (2); this reads off one partial-fraction term without contamination from the others. The method then interpolates the numerator n_k over several such points, repeats for each candidate denominator, and combines results across several primes via the Chinese remainder theorem to obtain the full exact rational function. The load-bearing feature is that most candidate terms vanish for loop-amplitude rational functions, so the one-at-a-time strategy detects and skips them cheaply.

What would settle it

Take a rational function whose exact partial-fraction decomposition includes a denominator not contained in the generated candidate list (for instance, a term $1/(x+y)$ while the candidate list only contains $x$ and $y$), run the p-adic one-term-at-a-time reconstruction, and compare the reconstructed expression with the exact one; any discrepancy shows the candidate-list assumption has failed.

Watch

Extended reading notes

Core claim

The central claim is that if one can list the candidate denominators that may appear in the partial-fractioned form of a rational function R, then R can be reconstructed term by term: choose p-adic points at which one candidate denominator d_k becomes p-adically smaller than all others, and the p-adic expansion of R(x) reads off n_k/d_k plus corrections of order 1/$p^{{m-1}}$. This lets each numerator n_k be interpolated separately, and—crucially—lets numerators that vanish be detected from a single probe and skipped. The paper reports that for the benchmark function R*, this strategy required roughly 6×$10^{4}$ p-adic probes per prime versus 1.4×$10^{6}$ finite-field probes, and produced a partial-fractioned expression with 52,527 fitted parameters (15,403 non-zero) instead of 1,369,559, reducing the size from 605 MB to 4.5 MB. The reconstructed expression was confirmed to be mathematically exactly equal, though far smaller. The paper also observes that 70% of the fitted parameters vanish, that some numerators are integer multiples of each other, and that some groups of terms recombine into simpler single terms, all features that could be exploited in future work.

Load-bearing premise

The method assumes that for the target rational function one can generate a complete list of candidate denominators for its partial-fractioned form, and that for each such denominator one can find p-adic points where it is p-adically smaller than all the others; the paper states the first is straightforward but does not prove either property for general rational functions.

Editorial extensions

If this is right

If the central claim is correct, multi-loop rational functions can be reconstructed from about 25 times fewer numerical probes than finite-field-based methods, easing a major computational bottleneck.
The reconstructed partial-fractioned expressions are roughly two orders of magnitude more compact than common-denominator forms, so downstream symbolic manipulations are much cheaper.
Vanishing partial-fraction terms—70% of the fitted parameters in the benchmark—are detected with a single p-adic probe each, meaning the savings grow as the simplification under partial fractioning grows.
The observed patterns (integer-multiple relations among numerators and recombinable groups of terms) indicate additional free-parameter reductions are available, potentially lowering probe counts further.
The method provides a concrete tool for studying why partial fractioning simplifies loop-amplitude rational functions so dramatically, since the reconstruction itself exposes the structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The one-term-at-a-time strategy could be extended to other rational functions that exhibit strong partial-fraction simplification, such as integration-by-parts reduction coefficients in other processes, where the same p-adic isolation condition may hold even though the paper does not prove it.
If the candidate-denominator list could be generated automatically by algebraic-geometry probes rather than manually, the method would become fully automated and could be applied to rational functions whose partial-fraction structure is not yet understood.
The observed integer-multiple relations between different numerators suggest hidden symmetries or group-theoretic structures in the amplitude; systematically searching for such relations before reconstruction could reduce the fitted-parameter count well below the reported 52,527.
The reported factor-of-25 probe reduction may be a lower bound: as the method is refined to exploit the zero numerators and linear relations, the savings for larger, more complicated functions could grow faster than for R*.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the method itself is in the author's own PRD [1]; this proceedings is a summary plus a demonstration on one large rational function R*. For what it is, it's a clean and honest write-up. The numbers are concrete and consistent: 605 MB down to 4.5 MB, 1,369,559 free parameters down to 52,527, and about 6e4 p-adic probes per prime versus 1.4e6 finite-field probes. The paper states the reconstructed expression was checked for exact equality, and the patterns in Sec. 3 (zero numerators, integer multiples, occasional combinability) are presented as post-hoc observations, not as exploited savings. That is appropriately cautious.

The soft spot is the one the stress-test flags, and I think it lands. The method rests on two oracles: a complete candidate-denominator list {d_i}, and a p-adic point that isolates each d_k as in Eq. (2). Neither is proved or even sketched here; both are deferred to [1]. Because multivariate partial fractions are not canonical, the completeness of that candidate list is genuinely nontrivial; the 1/(xy) example in the stress-test is a good illustration. The paper itself acknowledges that partial-fraction simplification is a special property of loop amplitudes, so the general claim is not established. For a conference proceedings, deferring to [1] is reasonable, but a reader cannot replay or independently assess the method from this text. No code or data artifacts are provided, so the single example rests on the author's assertion of an exact-equality check.

Also, the '25x fewer probes' counts p-adic probes against finite-field probes; the relative computational cost per probe is deferred to [1], so the practical speedup is not yet quantified here.

I would send this to a referee if it crossed my desk. The potential impact is real, and the demonstration on a challenging two-loop function is exactly the kind of stress test that matters. A referee should check the claims against [1], and ideally ask for the reconstructed expression and code to be released. But the referee should not accept the generality claims without proof of the candidate-denominator oracle. The paper is coherent and does not oversell; the softness is in what is omitted, not in what is said.

Referee Report

3 major / 4 minor

Summary. This conference proceedings paper describes a p-adic interpolation method for reconstructing rational functions directly in partial-fractioned form, term by term, using evaluations at specially chosen integer points. The method is demonstrated on R*, the largest rational function in an IBP expression used in the two-loop full-colour diphoton-plus-jet calculation of Ref. [10]. The paper reports that the reconstruction requires roughly 25 times fewer probes than conventional finite-field interpolation and yields an expression 134 times smaller by ByteCount, while the interpolated result is stated to be mathematically exactly equal to the original expression. The paper is explicitly a summary of the author's earlier article [H.A. Chawdhry, Phys. Rev. D 110, 056028 (2024)], to which all technical details are deferred.

Significance. If the method works as claimed, it is a practically important advance for multi-loop amplitude computations: partial fractioning is known to reduce the size of loop-amplitude rational functions by orders of magnitude, and reconstructing directly in that form avoids building and interpolating the enormous common-denominator expression. The benchmark R* is a realistic, non-trivial example from a state-of-the-art calculation, and the exact-equality check against the external IBP expression is a non-circular validation. The paper does not, however, establish the two key algorithmic prerequisites on which the method rests, and it presents no code, data artefacts, or a complete specification of the verification; these are load-bearing gaps for the reported savings.

major comments (3)

[§2, Eq. (2)] The central step of the method assumes (i) a complete list of candidate denominators {d_i} in the partial-fractioned form and (ii) for each non-vanishing d_k the existence of a point x in Q_p^N with |d_k(x)|_p < |d_i(x)|_p for all i ≠ k. The manuscript states that generating the candidate list is 'straight-forward' and defers the point-generation to Ref. [1], but gives no statement of the conditions under which these hold. In multivariate partial fractioning the decomposition is not canonical (e.g., 1/(xy) is not expressible as A/x + B/y with polynomials A and B), so completeness of the candidate list is not automatic. If the list is incomplete, or if no isolating point exists for some denominator, the leading p-adic term in Eq. (2) will mix contributions and the one-at-a-time reconstruction will misattribute or miss terms. This is load-bearing for the reported 25x probe reduction and the exactness claim; the paper should state these as precise assumptions, prove them for the relevant class of functions, or explicitly limit the claims to the case where they hold.
[§3, Table 1 and probe-count comparison] The comparison to 'conventional finite-field methods' counts only the number of probes: roughly 6*10^4 p-adic probes versus 1.4*10^6 finite-field probes. The paper does not discuss the relative computational cost per probe; p-adic evaluations are not claimed to be as cheap as finite-field modular arithmetic, and the practical significance of a 25x reduction in probe count depends on that cost. The paper says details of p-adic evaluation costs are in Ref. [1] (end of §2), but as a standalone account the claim 'can require 25 times fewer numerical probes' should be clearly labelled as a probe-count statement, not a runtime statement; if the authors intend a stronger computational-cost claim, the per-probe cost comparison must be included.
[§3, exactness check] The sentence 'we confirmed that the interpolated result ... is mathematically exactly equal to the original expression' is a key validation, but the procedure for this check is not described. Was the difference simplified symbolically, evaluated at further p-adic points, or compared via a polynomial identity test? Without this information, and without code or data artefacts, the reader cannot assess or replay the verification. Please specify the verification method at least briefly, for instance by stating the criterion used to certify exact equality.

minor comments (4)

[Abstract and §3] There are several typographical errors: 'calculatin g' in the abstract, 'expresssion' in §3, and 'belive' in §4. I recommend a careful proofread.
[Reference [21]] Reference [21] contains a garbled particle symbol: 'Wbb /macron.ts1' should presumably be 'W b-bar' or similar. Please correct the LaTeX source.
[Table 1] In Table 1, the 'Parameters to fit' entry is formatted as '52,527 ( of which 15,403 non-zero )' with an extra space after the parenthesis; please clean up the typography.
[§3, Eq. (3)] Equation (3) is visually dense and the notation such as 's6_45 s3_12' is ambiguous without explicit multiplication signs or parentheses. Please clarify the notation, possibly by using explicit products or a more compact display.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found; the central reconstruction is externally benchmarked and checked by exact equality, with technical prerequisites deferred to the author's prior article [1] as a non-circular support gap.

full rationale

The derivation is not circular. R* is reconstructed by fitting 52,527 candidate coefficients from p-adic evaluations of the explicit rational function and then checked for exact equality against the original expression from the independent IBP calculation of Ref. [10]: 'we confirmed that the interpolated result, although different in size, is mathematically exactly equal to the original expression.' Eq. (2) is a genuine p-adic ultrametric expansion rather than a tautological definition of n_k. The one caveat is in Sec. 2: the completeness of the candidate-denominator list and the existence of p-adic points isolating a single denominator are asserted and deferred to the author's own article ('For reasons explained in our main article [1], it is straight-forward to generate a list of candidate denominators {d_i}...' and 'It also includes explanations of how to generate points that pick out only one candidate denominator d_k'). This is load-bearing self-citation in the sense that these prerequisites are not re-derived in the proceedings, but it is not circular: Ref. [1] is a separate published article and the numerical benchmark and exact-equality check are external to the citation chain. The post-hoc patterns (70% zero parameters, n1 = 25 n2, recombination into Eq. 6) are explicitly not used in producing Table 1, so they cannot force the fitted result.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities. The central claim rests on p-adic valuation properties plus two domain-specific empirical assumptions: the candidate denominator list is generatable in advance, and loop-amplitude rational functions are unusually sparse in partial-fractioned form. The many coefficients fitted during interpolation are the output of the method, not free parameters introduced ad hoc by the paper.

assumptions (4)

standard math If one candidate denominator d_k is p-adically smaller than all other denominators at a p-adic point x, then the evaluation R(x) equals n_k/d_k up to higher-order terms, allowing isolated reconstruction of one partial-fraction term.
This is the non-Archimedean/ultrametric property of p-adic valuations used in Eq. (2). It is standard mathematics but is the key enabling mechanism for the one-term-at-a-time strategy.
domain assumption A complete list of candidate denominators {d_i} for the partial-fractioned form of R can be generated before reconstruction from easily available information such as mass dimension and the common denominator.
Stated in Sec. 2 as 'straight-forward to generate a list of candidate denominators'; if this list is incomplete, the reconstruction can silently miss terms.
domain assumption Rational functions that appear in multiloop scattering amplitude calculations simplify dramatically under partial fractioning, with many candidate numerators vanishing.
This is an empirical observation cited from recent literature in Sec. 1 and demonstrated on a single example, R*. It is not a generic property of rational functions, as the paper itself notes.
domain assumption The original expression for R* taken from Ref. [10], and its common denominator, are correct and usable as a benchmark.
The exact-equality check in Sec. 3 compares against this original expression; any error there would invalidate the verification of the interpolation result.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Rational-function interpolation from p-adic evaluations in scattering amplitude calculations." pith.science (2026). https://pith.science/paper/ILX2J7JK

@misc{pith2026241219868,
  author       = {Pith},
  title        = {Pith review of: Rational-function interpolation from p-adic evaluations in scattering amplitude calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILX2J7JK}},
  note         = {Machine review of arXiv:2412.19868}
}

read the original abstract

Numerical interpolation techniques are widely employed for calculating large rational functions in scattering amplitude computations. It has been observed in recent years that these rational functions greatly simplify upon partial fractioning. In this conference proceedings paper, based on the article [H. A. Chawdhry, Phys. Rev. D 110, 056028 (2024)], a technique is presented to interpolate such rational functions directly in partial-fractioned form, from evaluations at special integer points chosen for their properties under a p-adic absolute value. It is shown that the technique can require 25 times fewer numerical probes than conventional finite-field-based techniques and can produce results that are more compact in size by 2 orders of magnitude. The reconstructed results moreover exhibit additional patterns that could be exploited in future work to further improve the size of the results and the number of required numerical probes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 4 canonical work pages

[1]

Chawdhry, p-adic reconstruction of rational functions in multiloop a mplitudes, Phys

H.A. Chawdhry, p-adic reconstruction of rational functions in multiloop a mplitudes, Phys. Rev. D 110 (2024) 056028 [ 2312.03672]

arXiv 2024
[10]

Agarwal, F

B. Agarwal, F. Buccioni, A. von Manteuﬀel and L. Tancredi, Two-Loop Helicity Amplitudes for Diphoton Plus Jet Production in Full Color , Phys. Rev. Lett. 127 (2021) 262001 [ 2105.04585]

arXiv 2021
[2]

von Manteuﬀel and R.M

A. von Manteuﬀel and R.M. Schabinger, A novel approach to integration by parts reduction , Phys. Lett. B 744 (2015) 101 [ 1406.4513]

arXiv 2015
[3]

Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate fu nctional reconstruction, JHEP 12 (2016) 030 [ 1608.01902]

T. Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate fu nctional reconstruction, JHEP 12 (2016) 030 [ 1608.01902]

arXiv 2016
[4]

Klappert and F

J. Klappert and F. Lange, Reconstructing rational functions with FireFly , Comput. Phys. Commun. 247 (2020) 106951 [ 1904.00009]

arXiv 2020
[5]

Peraro, FiniteFlow: multivariate functional reconstruction usin g ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

T. Peraro, FiniteFlow: multivariate functional reconstruction usin g ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

arXiv 2019
[6]

Klappert, S.Y

J. Klappert, S.Y. Klein and F. Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly , Comput. Phys. Commun. 264 (2021) 107968 [ 2004.01463]

arXiv 2021
[7]

Abreu, J

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD , JHEP 05 (2019) 084 [ 1904.00945]

arXiv 2019

Show all 29 references

[8]

Boehm, M

J. Boehm, M. Wittmann, Z. Wu, Y. Xu and Y. Zhang, IBP reduction coeﬃcients made simple , JHEP 12 (2020) 054 [ 2008.13194]

2020 arXiv
[9]

Agarwal, F

B. Agarwal, F. Buccioni, A. von Manteuﬀel and L. Tancredi, Two-loop leading colour QCD corrections to q ¯q → γγg and qg → γγq , JHEP 04 (2021) 201 [ 2102.01820]

2021 arXiv
[11]

Bendle, J

D. Bendle, J. Boehm, M. Heymann, R. Ma, M. Rahn, L. Ristau et a l., pfd-parallel, a Singular/GPI-Space package for massively parallel multiv ariate partial fractioning , Comput. Phys. Commun. 294 (2024) 108942 [ 2104.06866]

2024 arXiv
[12]

Heller and A

M. Heller and A. von Manteuﬀel, MultivariateApart: Generalized partial fractions , Comput. Phys. Commun. 271 (2022) 108174 [ 2101.08283]

2022 arXiv
[13]

Chawdhry, M

H.A. Chawdhry, M. Czakon, A. Mitov and R. Poncelet, Two-loop leading-colour QCD helicity amplitudes for two-photon plus jet production at the LHC , JHEP 07 (2021) 164 [ 2103.04319]

2021 arXiv
[14]

Badger, M

S. Badger, M. Becchetti, E. Chaubey, R. Marzucca and F. Sar andrea, One-loop QCD helicity amplitudes for pp → ttj to O( ε2), JHEP 06 (2022) 066 [ 2201.12188]

2022 arXiv
[15]

Abreu, J

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page , Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD , Phys. Rev. Lett. 122 (2019) 082002 [ 1812.04586]

2019 arXiv
[16]

Laurentis and D

G. Laurentis and D. Ma ˆ ıtre,Extracting analytical one-loop amplitudes from numerical evaluations, JHEP 07 (2019) 123 [ 1904.04067]

2019 arXiv
[17]

De Laurentis and D

G. De Laurentis and D. Ma ˆ ıtre, Two-Loop Five-Parton Leading-Colour Finite Remainders in the Spinor-Helicity Formalism, JHEP 02 (2021) 016 [ 2010.14525]

2021 arXiv
[18]

Wang, A p-adic algorithm for univariate partial fractions , in Proceedings of the Fourth ACM Symposium on Symbolic and Algebraic Computation , SYMSAC ’81, (New York, NY, USA), p

P.S. Wang, A p-adic algorithm for univariate partial fractions , in Proceedings of the Fourth ACM Symposium on Symbolic and Algebraic Computation , SYMSAC ’81, (New York, NY, USA), p. 212–217, Association for Computing Machinery, 1981, DOI

1981
[19]

Bendle, J

D. Bendle, J. B¨ ohm, W. Decker, A. Georgoudis, F.-J. Pfreund t, M. Rahn et al., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Spa ce, JHEP 02 (2020) 079 [ 1908.04301]

2020 arXiv
[20]

Badger, C

S. Badger, C. Brønnum-Hansen, D. Chicherin, T. Gehrmann, H .B. Hartanto, J. Henn et al., Virtual QCD corrections to gluon-initiated diphoton plus j et production at hadron colliders , JHEP 11 (2021) 083 [ 2106.08664]

2021 arXiv
[21]

Badger, H.B

S. Badger, H.B. Hartanto and S. Zoia, Two-Loop QCD Corrections to Wbb /macron.ts1 Production at Hadron Colliders, Phys. Rev. Lett. 127 (2021) 012001 [ 2102.02516]

2021 arXiv
[22]

Badger, H.B

S. Badger, H.B. Hartanto, J. Kry´ s and S. Zoia, Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-qua rk pair at the LHC , JHEP 11 (2021) 012 [ 2107.14733]

2021 arXiv
[23]

Abreu, F

S. Abreu, F. Febres Cordero, H. Ita, M. Klinkert, B. Page and V. Sotnikov, Leading-color two-loop amplitudes for four partons and a W boson in QCD , JHEP 04 (2022) 042 [ 2110.07541]

2022 arXiv
[24]

De Laurentis and B

G. De Laurentis and B. Page, Ans¨ atze for scattering amplitudes from p-adic numbers and algebraic geometry, JHEP 12 (2022) 140 [ 2203.04269]

2022 arXiv
[25]

De Laurentis, H

G. De Laurentis, H. Ita, M. Klinkert and V. Sotnikov, Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Gluon Channel , 2311.10086

arXiv
[26]

De Laurentis, H

G. De Laurentis, H. Ita and V. Sotnikov, Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Quark Channels , 2311.18752

arXiv
[27]

Campbell, G

J.M. Campbell, G. De Laurentis and R.K. Ellis, Vector boson pair production at one loop: analytic results for the process qqℓℓℓ′ℓ ′ g, JHEP 07 (2022) 096 [ 2203.17170]

2022 arXiv
[28]

Tkachov, A theorem on analytical calculability of 4-loop renormaliz ation group functions , Phys

F.V. Tkachov, A theorem on analytical calculability of 4-loop renormaliz ation group functions , Phys. Lett. B 100 (1981) 65

1981
[29]

Chetyrkin and F.V

K.G. Chetyrkin and F.V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159

1981

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Chawdhry, p-adic reconstruction of rational functions in multiloop a mplitudes, Phys

H.A. Chawdhry, p-adic reconstruction of rational functions in multiloop a mplitudes, Phys. Rev. D 110 (2024) 056028 [ 2312.03672]

arXiv 2024

[2] [10]

Agarwal, F

B. Agarwal, F. Buccioni, A. von Manteuﬀel and L. Tancredi, Two-Loop Helicity Amplitudes for Diphoton Plus Jet Production in Full Color , Phys. Rev. Lett. 127 (2021) 262001 [ 2105.04585]

arXiv 2021

[3] [2]

von Manteuﬀel and R.M

A. von Manteuﬀel and R.M. Schabinger, A novel approach to integration by parts reduction , Phys. Lett. B 744 (2015) 101 [ 1406.4513]

arXiv 2015

[4] [3]

Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate fu nctional reconstruction, JHEP 12 (2016) 030 [ 1608.01902]

T. Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate fu nctional reconstruction, JHEP 12 (2016) 030 [ 1608.01902]

arXiv 2016

[5] [4]

Klappert and F

J. Klappert and F. Lange, Reconstructing rational functions with FireFly , Comput. Phys. Commun. 247 (2020) 106951 [ 1904.00009]

arXiv 2020

[6] [5]

Peraro, FiniteFlow: multivariate functional reconstruction usin g ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

T. Peraro, FiniteFlow: multivariate functional reconstruction usin g ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

arXiv 2019

[7] [6]

Klappert, S.Y

J. Klappert, S.Y. Klein and F. Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly , Comput. Phys. Commun. 264 (2021) 107968 [ 2004.01463]

arXiv 2021

[8] [7]

Abreu, J

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD , JHEP 05 (2019) 084 [ 1904.00945]

arXiv 2019