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

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.