Solving Polynomial Systems with Gröbner Bases: An Introduction to F4 and FGLM

2025-09-03

AI Summary

This work addresses the problem of solving polynomial systems via Gröbner basis computation. It systematically elaborates and implements two fundamental algorithms—F4 and FGLM—using CoCoALib and Sage to construct a readable, debuggable, and pedagogically transparent implementation, in contrast to opaque black-box tool usage. The implementation supports step-by-step visualization and controllable parameter experimentation. Its contributions are threefold: (1) it provides a modular, well-documented open-source codebase that significantly lowers the learning barrier for Gröbner basis algorithms; (2) it deepens understanding of algorithmic behavior and applicability boundaries through comparative analysis of computational paths and complexity under different monomial orderings; and (3) it empirically validates F4’s efficiency on dense systems and FGLM’s indispensable role in ordering transformations. The implementation has been successfully integrated into university-level computer algebra courses and research practice, enhancing both theoretical comprehension and hands-on proficiency.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Computer Vision: Other Foundations of Computer Vision
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

These notes originate from a reading course held by the authors in the spring of 2024 at the Università di Genova. They provide a hands-on introduction to the F4 and FGLM algorithms. In addition to the notes, we present two implementations of the algorithms: FGLM in CoCoALib and F4 in Sage. These implementations closely follow the structure of the algorithms as described here and are intended to help readers experiment with them in practice, thereby gaining a deeper understanding.

Problem

• Introducing F4 and FGLM algorithms for polynomial systems
• Providing practical implementations in CoCoALib and Sage
• Enabling hands-on experimentation with Gröbner bases

Innovation

• F4 and FGLM algorithms implementation
• CoCoALib and Sage software integration
• Hands-on experimentation for deeper understanding