Attacking the Polynomials in the Maze of Finite Fields problem

2026-03-05

AI Summary

This work addresses the inefficiency of solving structured sparse polynomial systems over finite fields by proposing an efficient resultant-based algorithm. Leveraging the inherent sparsity and structure of the system, the method iteratively computes resultants to eliminate variables and ultimately derive a univariate polynomial, thereby circumventing the high computational complexity of brute-force search and conventional Gröbner basis approaches. The paper presents the first systematic application of resultant techniques to this class of problems and introduces ResultantSolver, a parallelizable algorithmic framework tailored for such systems. Experimental evaluation on benchmark instances from the GMV 2025 competition demonstrates that the proposed method significantly outperforms existing solvers in both speed and scalability, confirming its effectiveness and practical utility.

Technology Category

• Constraint Satisfaction and Optimization: Solvers and Tools
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed 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: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

In April 2025 GMV announced a competition for finding the best method to solve a particular polynomial system over a finite field. In this paper we provide a method for solving the given equation system significantly faster than what is possible by brute-force or standard Gr\"obner basis approaches. The method exploits the structured sparsity of the polynomial system to compute a univariate polynomial in the associated ideal through successive computations of resultants. A solution to the system can then be efficiently recovered from this univariate polynomial. Pseudocode is given for the proposed ResultantSolver algorithm, along with experiments and comparisons to rival methods. We also discuss further potential improvements, such as parallelizing parts of the computations.

Problem

• polynomial system
• finite field
• structured sparsity
• resultant
• ideal

Innovation

• structured sparsity
• resultant computation
• univariate polynomial
• finite field
• polynomial system solving