Optimized Gr\"obner basis algorithms for maximal determinantal ideals and critical point computations

Given polynomials g and f₁,,fₚ, all in [x₁,,xₙ] for some field , we consider the problem of computing the critical points of the restriction of g to the variety defined by f₁=⋯=fₚ=0. These are defined by the simultaneous vanishing of the fᵢ's and all maximal minors of the Jacobian matrix associated to (g,f₁, …, fₚ). 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 F₅-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 F₅ algorithm and compare it to the best previously known bound for computing critical points using Gr\"obner bases.