On the arithmetic complexity of computing Gröbner bases of comaximal determinantal ideals

Abstract

Let $M$ be an $n\times n$ matrix of homogeneous linear forms over a field $\mathbb{k}$. If the ideal ${\mathcal{I}}_{n-2}(M)$ generated by minors of size $n-1$ is Cohen-Macaulay, then the Gulliksen-NegÃ¥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 Gröbner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gröbner 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 Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner 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 $n^{2\ddot{I}‰+3}$ asymptotically, where $2\le \ddot{I}‰\le 3$ is any complexity exponent for matrix multiplication over $\mathbb{k}$.

26 pages, 2 algorithms; updated remarks after Theorem 6.5