Differential Galois Theory and Integration

The SAGEX review on scattering amplitudes Chapter 4: Multi-loop Feynman integrals

“…Here we elaborate the most relevant approaches. Before one considers to solve such systems, one may also analyze them further as exemplified in [112] in order gain further insight or to find further relations among them.…”

Analytic integration methods in quantum field theory: an Introduction

“…General aspects on the solution of differential equation systems were summarized in Ref. [14], while aspects of differential Galois theory were discussed in [13]. In the solution of ordinary differential equations emerging in the context of Feynman diagrams holonomic integration often provides a powerful tool, see.…”

“…The topics of the workshop included both techniques to reduce the number of Feynman diagrams by physical relations, such as the integration by parts relations [5][6][7] as well as the mathematical methods to compute these integrals analytically. The latter include the method of generalized hypergeometric functions [8] and the general theory of contiguous relations [9], the methods of integer relations [10], guessing methods of one-dimensional quantities, hyperlogarithms [11], the solution of master-integrals using difference and differential equations [12][13][14][15], Risch algorithms on nested integrals and rationalization algorithms [16], holonomic integration [17], the multivalued Almkvist-Zeilberger algorithm [18], expansion by regions [19], elliptic integrals and related topics [20,21], cutting techniques [22], and special multi-leg applications [23][24][25]. In different precision calculations these technologies are applied.…”