On the arithmetic complexity of computing Gr\"obner bases of comaximal determinantal ideals

TL;DR

This paper analyzes the computational complexity of a specialized Gr"obner basis algorithm for determinantal ideals, providing bounds based on algebraic and combinatorial properties, and introduces conjecture-based asymptotic results.

Contribution

It establishes sharp complexity bounds for the DetGB algorithm applied to determinantal ideals, linking algebraic structures with computational complexity under conjectural assumptions.

Findings

The size of reduced grevlex Gr"obner bases grows at least as n^6 in the zero-dimensional case.

The complexity of DetGB is bounded above by n^{2ω+3}, with ω related to matrix multiplication complexity.

Results depend on conjectures similar to Fr"oberg's conjecture.

Abstract

Let $M$ be an $n\times n$ matrix of homogeneous linear forms over a field $\Bbbk$. If the ideal $\mathcal{I}_{n-2}(M)$ generated by minors of size $n-1$ is Cohen-Macaulay, then the Gulliksen-Neg{\aa}rd complex is a free resolution of $\mathcal{I}_{n-2}(M)$. It has recently been shown that by taking into account the syzygy modules for $\mathcal{I}_{n-2}(M)$ which can be obtained from this complex, one can derive a refined signature-based Gr\"obner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gr\"obner basis for $\mathcal{I}_{n-2}(M)$. In this paper, we establish sharp complexity bounds on DetGB. To accomplish this, we prove several results on the sizes of reduced grevlex Gr\"obner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fr\"oberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gr\"obner basis of $\mathcal{I}_{n-2}(M)$ is bounded from below by $n^{6}$ asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by $n^{2\omega+3}$ asymptotically, where $2\le\omega\le 3$ is any complexity exponent for matrix multiplication over $\Bbbk$.