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

Sriram Gopalakrishnan, Vincent Neiger, Mohab Safey El Din

TL;DR

This work addresses the exact computation of critical points for the restriction of a polynomial $g$ to the zero set of $F=(f_1,...,f_p)$ by studying the ideal $\mathcal{I}(g,F)=\langle F\rangle+I_{p}(\operatorname{jac}(g,F))$. It advances Gröbner basis methods by exploiting the first syzygies of maximal minors via the Eagon–Northcott complex and introducing a new $F_5$-type criterion tailored to these structured systems. A detailed complexity bound is derived under genericity assumptions akin to Fröberg-type semi-regularity, and the approach is compared with Lazard’s algorithm, showing polynomial gains and potential exponential improvements when fully exploiting the syzygy criterion. The results have practical impact for efficient exact critical point computation in polynomial optimization and related areas where determinantal structures arise. All results are stated with precise algebraic definitions and degree bounds, providing a pathway to faster implementations in computer algebra systems.$

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.

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

TL;DR

This work addresses the exact computation of critical points for the restriction of a polynomial to the zero set of by studying the ideal . It advances Gröbner basis methods by exploiting the first syzygies of maximal minors via the Eagon–Northcott complex and introducing a new -type criterion tailored to these structured systems. A detailed complexity bound is derived under genericity assumptions akin to Fröberg-type semi-regularity, and the approach is compared with Lazard’s algorithm, showing polynomial gains and potential exponential improvements when fully exploiting the syzygy criterion. The results have practical impact for efficient exact critical point computation in polynomial optimization and related areas where determinantal structures arise. All results are stated with precise algebraic definitions and degree bounds, providing a pathway to faster implementations in computer algebra systems.$

Abstract

Given polynomials and , all in for some field , we consider the problem of computing the critical points of the restriction of to the variety defined by . These are defined by the simultaneous vanishing of the 's and all maximal minors of the Jacobian matrix associated to . 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 -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 algorithm and compare it to the best previously known bound for computing critical points using Gröbner bases.
Paper Structure (12 sections, 21 theorems, 32 equations, 4 figures, 3 algorithms)

This paper contains 12 sections, 21 theorems, 32 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signature Gröbner bases
  4. The first syzygies of maximal minors
  5. The Eagon-Northcott complex
  6. Leading terms of syzygies
  7. Critical points
  8. Complexity analysis
  9. New complexity bound
  10. Comparison with Lazard's algorithm
  11. $p,n$ fixed, $d_0$ grows
  12. $p,d_0$ fixed, $n$ grows

Key Result

proposition 1

Eisenbud1995 Let $\mathcal{R}$ be a ring and let $\mathcal{M}$ be a finite free $\mathcal{R}$-module with basis $e_1,\dots, e_s$. Then for any integer $d\ge 1$, $\mathcal{M}^{\otimes d}$ is a free module of rank $s^{d}$ and the set $\{e_{i_1}\otimes\cdots\otimes e_{i_d}:1\le i_1,\dots,i_d\le s\}$ is

Figures (4)

  • Figure 1: Speedup of \ref{['alg:maxminorsgb']}. Top: $p=3$; bottom: $p=4$
  • Figure 2: Speedup of an algorithm which computes full-rank Macaulay matrices. Top: $p=3$; bottom: $p=4$.
  • Figure 3: Speedup of \ref{['alg:maxminorsgb']}. Top: $d_0=3$; bottom: $d_0=4$.
  • Figure 4: Speedup of an algorithm which computes full-rank Macaulay matrices. Top: $d_0=3$; bottom: $d_0=4$.

Theorems & Definitions (24)

  • proposition 1
  • proposition 2
  • theorem 1
  • theorem 2: Hilbert's syzygy theorem, Eisenbud1995
  • corollary 1
  • proposition 3: BrunsVetter1988
  • proposition 4
  • proposition 5: Syzygy Criterion, EderFaugere2016
  • theorem 3: $F_5$ Criterion, Faugere2002
  • theorem 4: EagonNorthcott1962, Eisenbud2005
  • ...and 14 more