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.

翻译：本文为一般零维情形下的grevlex Gröbner基计算提供了新的复杂度上界，该结果依赖于Moreno-Socías猜想。我们首先形式化了正则序列的一个性质，该性质蕴含了一个众所周知的经验结论，称之为“递增度性质”。随后，基于Moreno-Socías猜想，我们对F4算法中多项式对的选择机制给出了新的理解。此外，对于相关次数的一半情况，我们得到了给定次数的grevlex Gröbner基中元素数量的精确公式。结合这些结果，我们推导出了F4追踪器算法的精确复杂度公式及其在变量数趋于无穷时的渐近行为。这些结果将现有最优复杂度上界改进了变量数指数量级的因子。