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

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.