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 $\Bbbk$.

翻译：设$M$是域$\Bbbk$上由齐次线性形式构成的$n\times n$矩阵。若由$n-1$阶子式生成的理想$\mathcal{I}_{n-2}(M)$是Cohen-Macaulay的，则Gulliksen-Neg{\aa}rd复形是$\mathcal{I}_{n-2}(M)$的一个自由分解。最新研究表明，通过利用此复形中可获得的$\mathcal{I}_{n-2}(M)$的合冲模，可以推导出精炼的基于签名的Gröbner基算法DetGB，该算法在计算$\mathcal{I}_{n-2}(M)$的grevlex Gröbner基时避免了归约到零。本文建立了DetGB的严格复杂度界。为此，我们证明了关于逆字典序理想的约化grevlex Gröbner基规模的若干结论，据此得到了两个主要复杂度结果，这些结果依赖于类似Fröberg猜想的假设。第一个结果指出，在零维情形下，$\mathcal{I}_{n-2}(M)$的约化grevlex Gröbner基规模渐近下界为$n^{6}$。第二个结果同样针对零维情形，指出DetGB的复杂度渐近上界为$n^{2\omega+3}$，其中$2\le\omega\le 3$是$\Bbbk$上矩阵乘法的任意复杂度指数。