Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations

Given polynomials $g$ and $f_1,\dots ,f_p$, all in $\mathbb{k}[x_1,\dots ,x_n]$ for some field $\mathbb{k}$, 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 $(g,f_1,\dots ,f_p)$. 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 Gröbner 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 Gröbner bases.