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

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 SPR, a new Mathematica package for the division and elimination of variables from polynomial systems. SPR 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 SPR 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 SPR 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 SPR can be used to find previously unknown Landau singularities.
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Taxonomy
TopicsPolynomial and algebraic computation · Cryptography and Residue Arithmetic · Algebraic Geometry and Number Theory
