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Differential Galois Theory and Integration

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Anti-Differentiation and the Calculation of Feynman Amplitudes

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

Abstract

In this chapter, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in Dreyfus and Weil (Computing the Lie algebra of the differential Galois group: The reducible case, ArXiv 1904.07925 2019) on several examples to reduce the differential system. This will give information on potential algebraic relations between integrals.

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Notes

  1. 1.

    A function is D-finite when its is a solution of a linear differential equations with coefficients in k.

  2. 2.

    See https://dlmf.nist.gov/31.12 or [13]. The notation HeunC is the syntax in Maple.

  3. 3.

    An invertible n × n matrix U such that U′ = AU.

  4. 4.

    A Lie algebra is called algebraic when it is the Lie algebra of a linear algebraic group. When the M i are given, this can be computed, see [56], Section 3, or [1], Section 6.

  5. 5.

    For this reason, some authors, including ourselves, call the decomposition \(A=\sum _{i=1}^s a_i M_i\) a Wei-Norman decomposition of A.

  6. 6.

    The notation \(H(\overline {\mathbf {k}})\) denotes matrices whose entries are in \(\overline {\mathbf {k}}\) and satisfy all the equations defining the algebraic group H.

  7. 7.

    A field k is a \(\mathcal {C}^{1}\)-field when every non-constant homogeneous polynomial P over k has a non-trivial zero provided that the number of its variables is more than its degree. For example, \(\mathcal {C}(x)\) is a \(\mathcal {C}^{1}\)-field and any algebraic extension of a \(\mathcal {C}^{1}\)-field is a \(\mathcal {C}^{1}\)-field (Tsen’s theorem).

  8. 8.

    The reader may also find a pdf version athttp://www.unilim.fr/pages_perso/jacques-arthur.weil/DreyfusWeilReductionExamples.pdf.

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Acknowledgements

This project has received funding from the ANR project De Rerum Natura ANR-19-CE40-0018. We warmly thank the organizers of the conference Antidifferentiation and the Calculation of Feynman Amplitudes for stimulating exchanges and lectures. We specially thank the referee for many observations and precisions that have enhanced the quality of this text.

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Dreyfus, T., Weil, JA. (2021). Differential Galois Theory and Integration. 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_7

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