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Given polynomials g and f₁, , fₚ, all in ₁, , 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.
Gopalakrishnan et al. (Sun,) studied this question.