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

Let M be an n× n matrix of homogeneous linear forms over a field . If the ideal Iₙ₋₂(M) generated by minors of size $n-1$ is Cohen-Macaulay, then the Gulliksen-Neg{}rd complex is a free resolution of Iₙ₋₂(M). It has recently been shown that by taking into account the syzygy modules for Iₙ₋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 Iₙ₋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 Iₙ₋₂(M) is bounded from below by n⁶ asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by n2ω+3 asymptotically, where 2≤ω≤ 3 is any complexity exponent for matrix multiplication over .