Reconstructing Laurent expansion of rational functions using p-adic numbers

TL;DR

This paper introduces a new p-adic number-based method for efficiently reconstructing Laurent expansions of rational functions, which can handle multiple variables and reduce computational costs in complex calculations.

Contribution

The novel approach uses p-adic evaluations to reconstruct Laurent expansions, improving efficiency and enabling multi-variable handling compared to existing methods.

Findings

Reconstructs Laurent expansions from p-adic evaluations.

Reduces computational resources for initial expansion orders.

Applicable to simplifying Feynman integral reductions.

Abstract

We propose a novel method for reconstructing Laurent expansion of rational functions using $p$-adic numbers. By evaluating the rational functions in $p$-adic fields rather than finite fields, it is possible to probe the expansion coefficients simultaneously, enabling their reconstruction from a single set of evaluations. Compared with the reconstruction of the full expression, constructing the Laurent expansion to the first few orders significantly reduces the required computational resources. Our method can handle expansions with respect to more than one variables simultaneously. Among possible applications, we anticipate that our method can be used to simplify the integration-by-parts reduction of Feynman integrals in cutting-edge calculations.