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

Sriram Gopalakrishnan; Vincent Neiger; Mohab Safey El Din

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arXiv · 2402.07353

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

Sriram Gopalakrishnan·Vincent Neiger·Mohab Safey El Din

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Abstract

Given polynomials $g$ and $f_1,\dots,f_p$, all in $\Bbbk[x_1,\dots,x_n]$ for some field $\Bbbk$, we consider the problem of computing the critical points of the restriction of $g$ to the variety defined by $f_1=\cdots=f_p=0$. These are defined by the simultaneous vanishing of the $f_i$'s and all maximal minors of the Jacobian matrix associated to $(g,f_1, \ldots, f_p)$. 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 $F_5$-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 $F_5$ algorithm and compare it to the best previously known bound for computing critical points using Gröbner bases.

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BibTeX RIS

Sriram Gopalakrishnan, Vincent Neiger, Mohab Safey El Din. 2024-02-12. Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations. https://arxiv.org/abs/2402.07353

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