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 $\Bbbk$ 上齐次线性形式的 $n\times n$ 矩阵。若由 $n-1$ 阶子式生成的理想 $\mathcal{I}_{n-2}(M)$ 为科恩-麦考利理想，则古利克森-内高复形是 $\mathcal{I}_{n-2}(M)$ 的自由分解。近期研究表明，通过考虑此复形可获得的 $\mathcal{I}_{n-2}(M)$ 合冲模，可推导出改进的基于符号的格罗布纳基算法 DetGB，该算法在计算 $\mathcal{I}_{n-2}(M)$ 的 grevlex 格罗布纳基时避免归约至零。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约化 grevlex 格罗布纳基规模的多个结论，进而基于类似弗勒贝格猜想的假设得到了两个主要复杂度结果。第一个结果指出，在零维情形下，$\mathcal{I}_{n-2}(M)$ 的约化 grevlex 格罗布纳基规模渐近下界为 $n^{6}$。第二个结果同样针对零维情形，表明 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。