Sampling Polynomial Rational Remainders with SP$\mathbb{Q}$R: A new Package for Polynomial Division and Elimination

TL;DR

SP$\ ext{Q}$R is a Mathematica package that improves polynomial division and elimination by sampling over finite fields, significantly reducing computational complexity and enabling new insights into Feynman integrals.

Contribution

The paper introduces SP$\ ext{Q}$R, a novel package that enhances polynomial elimination techniques using finite field sampling, overcoming expression swell and enabling advanced Feynman integral analysis.

Findings

Reduces runtime and memory in polynomial computations by orders of magnitude.

Effectively overcomes expression swell in Gr"obner basis calculations.

Identifies previously unknown Landau singularities in Feynman integrals.

Abstract

We introduce SP$\mathbb{Q}$R, a new Mathematica package for the division and elimination of variables from polynomial systems. SP$\mathbb{Q}$R works by sampling and reconstructing results over finite fields, in an analogous manner to many state of the art Integration by Parts algorithms for Feynman integrals. This allows SP$\mathbb{Q}$R to effectively overcome expression swell during the construction of Gr\"obner bases, which in many cases is the major bottleneck in such computations. Benchmarks on state of the art Macaulay resultants show that SP$\mathbb{Q}$R can deliver substantial gains over symbolic computer algebra workflows -- reducing both runtime and memory footprint by multiple orders of magnitude. Likewise when applied to study Feynman integrals, we show how SP$\mathbb{Q}$R can be used to find previously unknown Landau singularities.