Refined $F_5$ Algorithms for Ideals of Minors of Square Matrices
Abstract
We consider the problem of computing a grevlex Gröbner 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 generalpurpose algorithm $F_5$ to find the sought Gröbner 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 $r=n2$, 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.05375
 arXiv:
 arXiv:2302.05375
 Bibcode:
 2023arXiv230205375G
 Keywords:

 Computer Science  Symbolic Computation;
 Mathematics  Commutative Algebra
 EPrint:
 21 pages, 3 algorithms