We consider the problem of computing a grevlex Gr\"obner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular sequences; in fact, the ideal $\langle F_r(M) \rangle$ cannot be generated by a regular sequence. As such, when using the general-purpose algorithm $F_5$ to find the sought Gr\"obner basis, some computing time is wasted on reductions to zero. We use known results about the first syzygy module of $F_r(M)$ to refine the $F_5$ algorithm in order to detect more reductions to zero. In practice, our approach avoids a significant number of reductions to zero. In particular, in the case $r=n-2$, we prove that our new algorithm avoids all reductions to zero, and we provide a corresponding complexity analysis which improves upon the previously known estimates.

翻译：我们考虑在特征为零或足够大的域上，计算 $n\times n$ 一般线性形式矩阵 $M$ 的 $r$ 阶子式集合 $F_r(M)$ 的 grevlex Gröbner 基问题。此类集合并非正则序列；事实上，理想 $\langle F_r(M) \rangle$ 无法由正则序列生成。因此，在使用通用算法 $F_5$ 求解目标 Gröbner 基时，部分计算时间会浪费在归零约简上。本文利用 $F_r(M)$ 的第一合冲模的已知结果对 $F_5$ 算法进行精炼，以检测更多归零约简。在实践中，我们的方法避免了大量归零约简。特别地，在 $r=n-2$ 情形下，我们证明新算法避免了所有归零约简，并给出了相应的复杂度分析，该分析改进了此前已知估计。