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Pré-Publication, Document De Travail Année : 2024

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

Vincent Neiger
Mohab Safey El Din

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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Dates et versions

hal-04450728 , version 1 (10-02-2024)
hal-04450728 , version 2 (26-07-2024)

Identifiants

  • HAL Id : hal-04450728 , version 1

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Sriram Gopalakrishnan, Vincent Neiger, Mohab Safey El Din. Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations. 2024. ⟨hal-04450728v1⟩
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