Elimination Templates in Macaulay2

2026-04-30

AI Summary

This work addresses the lack of systematic, automated solvers for families of zero-dimensional radical ideals involving algebraically independent parameters by introducing the EliminationTemplates package in Macaulay2. The package provides the first implementation within Macaulay2 of a general framework for constructing and specializing elimination templates, integrating elimination theory, Gröbner bases, and parameter specialization techniques. By extending solver methodologies originally developed in computer vision to broader algebraic contexts, it enables the creation of reusable, automated solvers. The effectiveness and practicality of this approach are demonstrated through successful applications to multiple computer vision problems as well as other algebraic scenarios.

Abstract

We introduce the package \texttt{EliminationTemplates} for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. This article provides a self-contained description of how elimination templates are constructed for such families and their specialization properties. Additionally, we describe the main functionality and datatypes provided by our package, and illustrate its usage on several examples, including applications from computer vision from which elimination templates originated.

Problem

• elimination templates
• zero-dimensional radical ideals
• algebraically independent parameters
• automatic solvers
• computer algebra

Innovation

• Elimination Templates
• Macaulay2
• zero-dimensional radical ideals
• algebraic parameters
• automatic solvers