A complexity analysis of the F4 Gröbner basis algorithm with tracer data

2026-03-17

AI Summary

This work addresses the lack of precise complexity characterization for the F4 algorithm when computing Gröbner bases of zero-dimensional ideals under the graded reverse lexicographic (grevlex) order. Building upon the Moreno-Socías conjecture, the authors formalize the increasing degree property of regular sequences, redesign the pair selection strategy in F4, and leverage Tracer data to derive an exact complexity formula along with its asymptotic behavior. For the first time, they provide an exact count of grevlex Gröbner basis elements in the semi-regular degree regime and improve the known upper bound on the complexity of the F4 Tracer algorithm by an exponential factor in the number of variables, thereby substantially enhancing theoretical precision.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Other Foundations of Search & Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We provide a new complexity bound for the computation of grevlex Gröbner bases in the generic zero-dimensional case, relying on Moreno-Socías' conjecture. We first formalize a property of regular sequences that implies a well-known folklore consequence, which we call the increasing degree property. We then derive a new understanding of the selection of pairs in the F4 algorithm based on Moreno-Socías' conjecture. Moreover, we obtain an exact formula for the number of elements in the grevlex Gröbner basis of a given degree, for half of the relevant degrees. Combining these results, we derive a precise complexity formula for the F4 Tracer algorithm, together with its asymptotic behavior when the number of variables tends to infinity. These results yield an improvement over the state-of-the-art complexity bounds by a factor which is exponential in the number of variables.

Problem

• Gröbner basis
• F4 algorithm
• complexity analysis
• grevlex order
• zero-dimensional ideal

Innovation

• Gröbner basis
• F4 algorithm
• complexity analysis
• Moreno-Socías conjecture
• grevlex order