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Taming symbolic integration-by-parts reduction with intermediate bases

Qian Song*

  • *Contact author: qian.song@ugent.be
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Phys. Rev. D 114, 096004 – Published 4 November, 2026

DOI: https://doi.org/10.1103/34kt-s9vz

Abstract

Despite many years of development in integration-by-parts reduction, reconstructing all reduction coefficients, which are rational functions of kinematic variables and the space-time dimension, remains a nontrivial problem. The main difficulty comes from the large number of unknowns in a general ansatz, which can lead to a linear system that is too large to solve. In this paper, we present an algorithm for reconstructing reduction coefficients through a sequence of intermediate bases. The resulting analytic reduction coefficients are products of a few analytic matrices, whose nonzero entries are simple rational functions. We demonstrate the efficiency of this algorithm with two cutting edge examples; a three-point massive box-triangle and a four-point massive pentagon-triangle. Reconstructing all reduction coefficients for the box-triangle (pentagon-triangle) requires 3289 (13, 013) numerical samplings, significantly fewer than the number of unknowns in the general ansatz, 1, 407, 406 (21, 638, 331).

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References (50)

  1. K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B192, 159 (1981).
  2. S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
  3. Charalampos Anastasiou and Achilleas Lazopoulos, Automatic integral reduction for higher order perturbative calculations, J. High Energy Phys. 07 (2004) 046.
  4. R. N. Lee, Presenting LiteRed: A tool for the Loop InTEgrals REDuction, arXiv:1212.2685.
  5. Roman N. Lee, LiteRed 1.4: A powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523, 012059 (2014).
  6. A. V. Smirnov, Algorithm FIRE—Feynman Integral REduction, J. High Energy Phys. 10 (2008) 107.
  7. A. V. Smirnov and V. A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184, 2820 (2013).
  8. Alexander V. Smirnov, FIRE5: A C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189, 182 (2015).
  9. A. V. Smirnov and F. S. Chukharev, FIRE6: Feynman Integral REduction with modular arithmetic, Comput. Phys. Commun. 247, 106877 (2020).
  10. C. Studerus, Reduze—Feynman integral reduction in C++, Comput. Phys. Commun. 181, 1293 (2010).
  11. A. von Manteuffel and C. Studerus, Reduze 2—Distributed Feynman Integral Reduction, arXiv:1201.4330.
  12. Philipp Maierhöfer, Johann Usovitsch, and Peter Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun. 230, 99 (2018).
  13. Jonas Klappert, Fabian Lange, Philipp Maierhöfer, and Johann Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
  14. Zihao Wu, Janko Boehm, Rourou Ma, Hefeng Xu, and Yang Zhang, NeatIBP 1.0, a package generating small-size integration-by-parts relations for Feynman integrals, Comput. Phys. Commun. 295, 108999 (2024).
  15. Alexander V. Smirnov and Mao Zeng, Feynman integral reduction: Balanced reconstruction of sparse rational functions and implementation on supercomputers in a co-design approach, Numer. Methods Program. 25, 30 (2024).
  16. Andreas von Manteuffel and Robert M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744, 101 (2015).
  17. Tiziano Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, J. High Energy Phys. 12 (2016) 030.
  18. Jonas Klappert and Fabian Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247, 106951 (2020).
  19. Tiziano Peraro, FiniteFlow: Multivariate functional reconstruction using finite fields and dataflow graphs, J. High Energy Phys. 07 (2019) 031.
  20. Andreas Maier, Scaling up to multivariate rational function reconstruction, Comput. Phys. Commun. 317, 109827 (2025).
  21. A. V. Belitsky, A. V. Smirnov, and R. V. Yakovlev, Balancing act: Multivariate rational reconstruction for IBP, Nucl. Phys. B993, 116253 (2023).
  22. Michael Monagan, Maximal quotient rational reconstruction: An almost optimal algorithm for rational reconstruction, in Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, ISSAC ’04 (Association for Computing Machinery, New York, USA, 2004), pp. 243–249, 10.1145/1005285.1005321.
  23. Michael Ben-Or and Prasoon Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation, in Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC ’88 (Association for Computing Machinery, New York, USA, 1988), pp. 301–309, 10.1145/62212.62241.
  24. R. Zippel, Interpolating polynomials from their values, J. Symb. Comput. 9, 375 (1990).
  25. D. Grigoriev, M. Karpinski, and M. F. Singer, Computational complexity of sparse rational interpolation, SIAM J. Comput. 23, 1 (1994).
  26. Erich Kaltofen, Greatest common divisors of polynomials given by straight-line programs, J. ACM 35, 231 (1988).
  27. E. Kaltofen and B. M. Trager, Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators, J. Symb. Comput. 9, 301 (1990).
  28. Erich Kaltofen and Zhengfeng Yang, On exact and approximate interpolation of sparse rational functions, in Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, ISSAC ’07 (Association for Computing Machinery, New York, USA, 2007), pp. 203–210, 10.1145/1277548.1277577.
  29. Jennifer de Kleine, Michael Monagan, and Allan Wittkopf, Algorithms for the non-monic case of the sparse modular gcd algorithm, in Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ISSAC ’05 (Association for Computing Machinery, New York, USA, 2005), pp. 124–131, 10.1145/1073884.1073903.
  30. Angel Díaz and Erich Kaltofen, Foxbox: A system for manipulating symbolic objects in black box representation, in Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ISSAC ’98 (Association for Computing Machinery, New York, USA, 1998), pp. 30–37, 10.1145/281508.281538.
  31. Erich Kaltofen, Wen-shin Lee, and Austin A. Lobo, Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel’s algorithm, in Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ISSAC ’00 (Association for Computing Machinery, New York, USA, 2000), pp. 192–201, 10.1145/345542.345629.
  32. Qiao-Long Huang and Xiao-Shan Gao, Sparse polynomial interpolation with finitely many values for the coefficients, arXiv:1704.04359.
  33. Jonas Klappert, Sven Yannick Klein, and Fabian Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly, Comput. Phys. Commun. 264, 107968 (2021).
  34. Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107, 315 (2007).
  35. Johannes M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013).
  36. Roman N. Lee, Reducing differential equations for multiloop master integrals, J. High Energy Phys. 04 (2015) 108.
  37. Xiao Liu, Reconstruction of rational functions made simple, Phys. Lett. B 850, 138491 (2024).
  38. A. V. Smirnov and V. A. Smirnov, How to choose master integrals, Nucl. Phys. B960, 115213 (2020).
  39. Johann Usovitsch, Factorization of denominators in integration-by-parts reductions, arXiv:2002.08173.
  40. S. Abreu, J. Dormans, F. F. Cordero, H. Ita, and B. Page, Analytic form of planar two-loop five-gluon scattering amplitudes in QCD, Phys. Rev. Lett. 122, 082002 (2019).
  41. Giuseppe De Laurentis and Ben Page, Ansätze for scattering amplitudes from p-adic numbers and algebraic geometry, J. High Energy Phys. 12 (2022) 140.
  42. Herschel A. Chawdhry, p-adic reconstruction of rational functions in multiloop amplitudes, Phys. Rev. D 110, 056028 (2024).
  43. Alexander V. Smirnov, Vladislav. A. Fokin, and Egor Yu. Chuvashov, Factorization of denominators as a “fuel” for Feynman integral reduction, arXiv:2605.13256.
  44. Ben Page and Qian Song, Critical points and syzygies for Feynman integrals, J. High Energy Phys. 02 (2026) 004.
  45. Sid Smith and Mao Zeng, Feynman integral reduction using Syzygy-constrained symbolic reduction rules, J. High Energy Phys. 01 (2026) 102.
  46. Leonardo de la Cruz and David A. Kosower, Seedless reduction of Feynman integrals, arXiv:2602.22111.

  47. Such an estimation is based on evaluation on 32-bit prime number with private linear solver.

  48. Philipp Maierhöfer and Johann Usovitsch, Kira 1.2 Release Notes, arXiv:1812.01491.
  49. Fabian Lange, Johann Usovitsch, and Zihao Wu, Kira 3: Integral reduction with efficient seeding and optimized equation selection, Comput. Phys. Commun. 322, 109999 (2026).

  50. For our interested part of physical amplitude, O(ε2) is enough.

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Phys. Rev. D 114, 096004– Published 4 November, 2026
  • Received 13 August 2026
  • Accepted 14 October 2026
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