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

Vsevolod Chestnov, Giulio Crisanti

TL;DR

The paper introduces SP$\mathbb{Q}$R, a Mathematica package that performs polynomial division and variable elimination through finite-field sampling and reconstruction, thereby avoiding intermediate expression swell common in Gröbner-basis methods. By leveraging companion matrices, elimination theory, and a FiniteFlow back end, SP$\mathbb{Q}$R achieves substantial speedups and memory reductions on parameter-rich problems, notably in Macaulay resultants and Landau analysis for Feynman integrals. The authors provide a detailed theoretical and implementation framework, practical installation and usage guidance, and demonstrations showing new Landau singularities and competitive performance against state-of-the-art CAS. The work highlights when this finite-field, linear-algebra–centric approach is advantageous and outlines future improvements, including tighter integration with numerical Gröbner-basis procedures and enhanced reconstruction techniques. SP$\mathbb{Q}$R thus offers a scalable, parametric-aware toolkit for polynomial algebra with broad applicability in mathematics and high-energy physics.

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 Gröbner 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.

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

TL;DR

The paper introduces SPR, a Mathematica package that performs polynomial division and variable elimination through finite-field sampling and reconstruction, thereby avoiding intermediate expression swell common in Gröbner-basis methods. By leveraging companion matrices, elimination theory, and a FiniteFlow back end, SPR achieves substantial speedups and memory reductions on parameter-rich problems, notably in Macaulay resultants and Landau analysis for Feynman integrals. The authors provide a detailed theoretical and implementation framework, practical installation and usage guidance, and demonstrations showing new Landau singularities and competitive performance against state-of-the-art CAS. The work highlights when this finite-field, linear-algebra–centric approach is advantageous and outlines future improvements, including tighter integration with numerical Gröbner-basis procedures and enhanced reconstruction techniques. SPR thus offers a scalable, parametric-aware toolkit for polynomial algebra with broad applicability in mathematics and high-energy physics.

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 Gröbner 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.

Paper Structure

This paper contains 60 sections, 76 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Review of Polynomial Division
  4. Univariate Polynomial Division
  5. Univariate Rational Function Division
  6. Univariate polynomial division as linear system solving
  7. Vanishing sets and polynomial remainders
  8. Multivariate Polynomial Division
  9. Gröbner Bases
  10. Multivariate Polynomial Division as Linear System Solving
  11. Companion Matrices
  12. Basics of the Formalism
  13. Rational Function Reduction
  14. Roots and the Eigenvalue Theorem
  15. Elimination theory
  16. ...and 45 more sections

Figures (8)

  • Figure 1: The left panel shows the vanishing set of the ideal \ref{['eq:ideal_2var_explicit']}, while the right one depicts that of its Gröbner basis \ref{['eq:multivar_example_gb']}; the two sets coincide.
  • Figure 2: Projections of the root system from \ref{['eq:elim_2var_explicit']} obtained by eliminating the variables $y$ (left) and $x$ (right).
  • Figure 3: A flowchart showing the various stages in a computation in SP$\mathbb{Q}$R, where the various FiniteFlow commands used as a back end have been labelled.
  • Figure 4: Structure of non-zero elements of the Macaulay system from \ref{['eq:system_generation_markandsweep']}. The FFAlgTake function of FiniteFlow rearranges the coefficiants of the input polynomials $f_1$ and $f_2$ (shown in the top row) into the non-zero entries of the sparse Macaulay matrix, stored in row-major order.
  • Figure 5: Flowcharts illustrating two matrix operations in SP$\mathbb{Q}$R: a simple recursive algorithm that computes the $n^{\text{th}}$ matrix power (\ref{['fig:MatPower']}), and a subroutine for matrix inversion using a linear solver (\ref{['fig:MatInverse']}).
  • ...and 3 more figures