Abstract
Complex algebraic calculations can be performed by reconstructing analytic
results from numerical evaluations over finite fields. We describe FiniteFlow,
a framework for defining and executing numerical algorithms over finite fields
and reconstructing multivariate rational functions. The framework employs
computational graphs, known as dataflow graphs, to combine basic building
blocks into complex algorithms. This allows to easily implement a wide range of
methods over finite fields in high-level languages and computer algebra
systems, without being concerned with the low-level details of the numerical
implementation. This approach sidesteps the appearance of large intermediate
expressions and can be massively parallelized. We present applications to the
calculation of multi-loop scattering amplitudes, including the reduction via
integration-by-parts identities to master integrals or special functions, the
computation of differential equations for Feynman integrals, multi-loop
integrand reduction, the decomposition of amplitudes into form factors, and the
derivation of integrable symbols from a known alphabet. We also release a
proof-of-concept C++ implementation of this framework, with a high-level
interface in Mathematica.
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