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，为$n\times n$泛型线性形式矩阵$M$的$r$阶子式集合$F_r(M)$计算grevlex Gröbner基的问题。此类集合并非正则序列；事实上，理想$\langle F_r(M) \rangle$不能由正则序列生成。因此，当使用通用算法$F_5$求解目标Gröbner基时，部分计算时间会浪费在归约至零上。我们利用$F_r(M)$第一合冲模的已知结果来优化$F_5$算法，从而检测更多的归约至零情形。实践表明，我们的方法显著避免了大量归约至零操作。特别地，在$r=n-2$的情况下，我们证明新算法完全避免了所有归约至零，并给出相应的复杂度分析，该分析改进了此前已知的估计结果。