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Reconstructing Laurent expansion of rational functions using p-adic numbers

Tianya Xia, Li Lin Yang

TL;DR

The paper addresses reconstructing Laurent expansions of rational functions from $p$-adic evaluations to avoid full-expression reconstruction in high-precision IBP reductions. It builds on finite-field reconstruction, then leverages $p$-adic data to recover coefficients order-by-order, potentially for multiple variables, while exploiting denominator recycling to boost efficiency. Precision considerations in $p$-adic arithmetic are discussed, with strategies to mitigate carries and ensure reliable extraction of coefficients. Benchmarks in IBP contexts show dramatically fewer probes needed for leading orders, with competitiveness in per-probe cost relative to finite-field methods. Overall, the approach offers a practical, scalable path to obtain early-term Laurent coefficients and could streamline complex Feynman-integral calculations.

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.

Reconstructing Laurent expansion of rational functions using p-adic numbers

TL;DR

The paper addresses reconstructing Laurent expansions of rational functions from -adic evaluations to avoid full-expression reconstruction in high-precision IBP reductions. It builds on finite-field reconstruction, then leverages -adic data to recover coefficients order-by-order, potentially for multiple variables, while exploiting denominator recycling to boost efficiency. Precision considerations in -adic arithmetic are discussed, with strategies to mitigate carries and ensure reliable extraction of coefficients. Benchmarks in IBP contexts show dramatically fewer probes needed for leading orders, with competitiveness in per-probe cost relative to finite-field methods. Overall, the approach offers a practical, scalable path to obtain early-term Laurent coefficients and could streamline complex Feynman-integral calculations.

Abstract

We propose a novel method for reconstructing Laurent expansion of rational functions using -adic numbers. By evaluating the rational functions in -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.

Paper Structure

This paper contains 10 sections, 1 theorem, 27 equations, 1 figure, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basics of p-adic numbers
  3. Reconstruction of Laurent expansion of rational functions
  4. Finite field reconstruction of rational functions
  5. Laurent expansion with respect to a single variable
  6. Laurent expansion with respect to more than one variables
  7. Precision of p-adic numbers in computer systems
  8. Recycling information in the reconstruction of different orders
  9. Benchmarks in the reconstruction of IBP coefficients
  10. Summary

Key Result

Lemma 1

Let positive integers $a, b \in \mathbb{Z}_{>0}$ be coprime. Consider the equation $n = a x + b y$ with integer $n \in [0, 2ab - a - b]$. If there exist a solution $(x,y)$ in the range $0 \leq x < b$ and $0 \leq y < a$, then it is the unique solution in the range $x \geq 0$ and $y \ge 0$.

Figures (1)

  • Figure 1: The time consumption per probe in the finite field and in the $p$-adic field. The dashed lines show the time consumption in the finite field using two different interfaces fq.h and fq_nmod.h in FLINT. The solid lines represent the time consumption in the $p$-adic field as a function of the precision, with different variables set to $p$. The upper and lower plots correspond to the evaluation of a 400 KB expression and a 1.6 MB expression, respectively

Theorems & Definitions (1)

  • Lemma 1