Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations
Résumé
Given polynomials g and f1,…,fp, all in k[x1,…,xn] for some field k, 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.
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