Refined $F_5$ Algorithms for Ideals of Minors of Square Matrices

TL;DR

This paper improves the $F_5$ algorithm for computing Gröbner bases of minors of matrices by reducing unnecessary reductions to zero, especially for the case of minors of size $n-2$, leading to efficiency gains.

Contribution

The authors refine the $F_5$ algorithm using syzygy module results to detect more reductions to zero, significantly enhancing computational efficiency for ideals of minors.

Findings

New algorithm avoids all reductions to zero for minors of size n-2.

Complexity analysis shows improved performance over previous estimates.

Practical implementation reduces computation time in Gröbner basis calculations.

Abstract

We consider the problem of computing a grevlex Gr\"obner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular sequences; in fact, the ideal $\langle F_r(M) \rangle$ cannot be generated by a regular sequence. As such, when using the general-purpose algorithm $F_5$ to find the sought Gr\"obner basis, some computing time is wasted on reductions to zero. We use known results about the first syzygy module of $F_r(M)$ to refine the $F_5$ algorithm in order to detect more reductions to zero. In practice, our approach avoids a significant number of reductions to zero. In particular, in the case $r=n-2$, we prove that our new algorithm avoids all reductions to zero, and we provide a corresponding complexity analysis which improves upon the previously known estimates.