A complexity analysis of the F4 Gr\"obner basis algorithm with tracer data

TL;DR

This paper derives a new, more precise complexity bound for the F4 Gr"obner basis algorithm in the generic zero-dimensional case, using Moreno-Socías' conjecture and tracer data, showing exponential improvements over previous bounds.

Contribution

It introduces a new complexity analysis of the F4 algorithm based on Moreno-Socías' conjecture, providing exact formulas and asymptotic behavior for the basis computation.

Findings

New complexity bound for F4 algorithm

Exact formula for basis elements in certain degrees

Asymptotic analysis showing exponential improvement

Abstract

We provide a new complexity bound for the computation of grevlex Gr\"obner bases in the generic zero-dimensional case, relying on Moreno-Soc\'ias' 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-Soc\'ias' conjecture. Moreover, we obtain an exact formula for the number of elements in the grevlex Gr\"obner 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.