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

Given polynomials and 1 , . . ., , all in k[ 1 , . . ., ] for some field k, we consider the problem of computing the critical points of the restriction of to the variety defined by 1 = • • • = = 0.These are defined by the simultaneous vanishing of the 's and all maximal minors of the Jacobian matrix associated to (, 1 , . . ., ).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 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 5 algorithm and compare it to the best previously known bound for computing critical points using Gröbner bases.