Given polynomials $g$ and $f_1,\dots,f_p$, all in $\Bbbk[x_1,\dots,x_n]$ for some field $\Bbbk$, we consider the problem of computing the critical points of the restriction of $g$ to the variety defined by $f_1=\cdots=f_p=0$. These are defined by the simultaneous vanishing of the $f_i$'s and all maximal minors of the Jacobian matrix associated to $(g,f_1, \ldots, f_p)$. We use the Eagon-Northcott complex associated to the ideal generated by these maximal minors to gain insight into the syzygy module of the system defining these critical points. We devise new $F_5$-type criteria to predict and avoid more reductions to zero when computing a Gr\"obner basis for the defining system of this critical locus. We give a bound for the arithmetic complexity of this enhanced $F_5$ algorithm and compare it to the best previously known bound for computing critical points using Gr\"obner bases.

翻译：给定域$\Bbbk$上的多项式$g$和$f_1,\dots,f_p$（均在$\Bbbk[x_1,\dots,x_n]$中），我们考虑计算$g$在由$f_1=\cdots=f_p=0$所定义代数簇上的限制的临界点问题。这些临界点由$f_i$的零点以及$(g,f_1,\ldots,f_p)$的雅可比矩阵的所有最大子式的同时消去所定义。我们利用这些最大子式生成理想的Eagon-Northcott复形，深入分析定义该临界点系统的合冲模结构。针对该临界轨迹的定义系统，我们设计了新的$F_5$-型准则，用于预测并避免计算Gröbner基时更多的归零约化。我们给出了该增强型$F_5$算法的算术复杂度界，并与此前已知的最优Gröbner基临界点计算复杂度界进行了比较。