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

Sriram Gopalakrishnan (1, 2) , Vincent Neiger (2) , Mohab Safey El Din (2)

1 University of Waterloo [Waterloo]
2 PolSys - Polynomial Systems

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 Gr\"obner 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 Gr\"obner bases.

Mots clés: Gröbner bases; critical points; optimization; real algebraic geometry

Domaines: Informatique [cs]; Mathématiques [math]

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