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Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

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Anti-Differentiation and the Calculation of Feynman Amplitudes

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing the available difference ring theory. Based on that, precise input-output specifications of the available tools of the summation package Sigma are provided.

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Notes

  1. 1.

    For a ring we denote by the set of units. If is a field, this means .

  2. 2.

    We also write p instead of f ⊙ p if f = 1; similarly we write f instead of f ⊙ p if p = 1.

  3. 3.

    The sum-product reduced form is not necessary, but simplifies the proof given on page 25.

  4. 4.

    In general, we might need a larger field or where the field is extended to .

  5. 5.

    Note that is a difference ring extension of .

  6. 6.

    If t is an A-monomial, we have \(\operatorname {ev}(t,n)=\alpha ^n\).

  7. 7.

    This step is not necessary for the proof, but avoids unneccesary copies of APS-monomials. When we refine this construction later, this step will be highly relevant.

  8. 8.

    Here we get or where is a field extension of ; if , one can restrict to the special case .

  9. 9.

    For an alternative algorithm we refer to [118, Section 6].

  10. 10.

    If the extension is basic, we only need the case k = 1.

  11. 11.

    Since W is depth-optimal, it follows in particular that d(G′) ≤d(G).

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Acknowledgements

This article got inspired during the workshop Antidifferentiation and the Calculation of Feynman Amplitudes at Zeuthen, Germany. In this regard, I would like to thank DESY for its hospitality and the Wolfgang Pauli Centre (WPC) for the financial support. In particular, I would like to thank the referee for the very thorough reading and valuable suggestions which helped me to improve the presentation.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement No. 764850, SAGEX and from the Austrian Science Foundation (FWF) grant SFB F50 (F5005-N15) in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.

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Schneider, C. (2021). Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_17

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