# FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs

## TL;DR

FiniteFlow is a framework that uses finite fields and dataflow graphs to efficiently perform complex algebraic calculations, enabling advanced multi-loop scattering amplitude computations with high parallelization.

## Contribution

It introduces a high-level, flexible framework for multivariate rational function reconstruction over finite fields using computational graphs, simplifying complex algebraic calculations.

## Key findings

- Successfully applied to multi-loop scattering amplitude calculations
- Allows high-level implementation without low-level numerical details
- Supports massively parallel computations

## 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.