Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations
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| Format: | Preprint |
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2024
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| author | Gopalakrishnan, Sriram Neiger, Vincent Din, Mohab Safey El |
| author_facet | Gopalakrishnan, Sriram Neiger, Vincent Din, Mohab Safey El |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_07353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations Gopalakrishnan, Sriram Neiger, Vincent Din, Mohab Safey El Symbolic Computation Commutative Algebra 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. |
| title | Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations |
| topic | Symbolic Computation Commutative Algebra |
| url | https://arxiv.org/abs/2402.07353 |