AI 中文总结
本研究提出一种基于Klein和Reuter（STOC 2025）格基计算技术的算法，可将矩阵行列式计算的位复杂度优化为$\tilde{O}(d^{\omega(2)}\log\\|B\\|)$，比现有最快确定性算法快约$d^{0.1213}$倍。
AI 中文摘要
我们提出了一种具有自然几何解释的算法，用于计算矩阵$B\in\mathbb{Z}^{d\times d}$的行列式。该算法比当前最快的确定性算法快$d^{\omega(1)+1-\omega(2)}\approx d^{0.1213}$倍，其中$\omega(k)$表示计算$d\times d$矩阵与$d\times d^k$矩阵相乘所需的指数。我们的方法基于Klein和Reuter（STOC 2025）的最新成果，他们提出了一种用于格基计算的新颖算法思路，可视为将欧几里得算法从$\mathbb{Z}$扩展到$\mathbb{Z}^d$。通过适配他们的技术，我们计算行列式的位复杂度与对输入矩阵$A\in\mathbb{Z}^{d\times 2d}$应用广义欧几里得算法（其中$\\|A\\| = \\|B\\|$）相同，即$\tilde{O}(d^{\omega(2)}\log\\|B\\|)$。在本研究之前，最快的行列式计算确定性算法需要$\tilde{O}(d^{\omega(1)+1}\log\\|B\\|)$位操作。
英文摘要
We present an algorithm with a natural geometric interpretation for computing the determinant of a matrix $B\in\mathbb{Z}^{d\times d}$. It improves upon the current fastest deterministic algorithms by a factor of $d^{ω(1)+1-ω(2)}\approx d^{0.1213}$, where $ω(k)$ denotes the exponent required for multiplying a $d\times d$ matrix with a $d\times d^k$ matrix. Our approach builds on a recent result of Klein and Reuter (STOC 2025), who introduced a novel algorithmic idea for lattice basis computation that can be viewed as extending the Euclidean algorithm from $\mathbb{Z}$ to $\mathbb{Z}^d$. By adapting their techniques, we compute the determinant with the same bit complexity as applying the generalized Euclidean algorithm to an input matrix $A\in\mathbb{Z}^{d\times 2d}$ with $\|A\| = \|B\|$, namely $\tilde{O}(d^{ω(2)}\log\|B\|)$. Prior to this work, the fastest deterministic algorithm for computing the determinant required $\tilde{O}(d^{ω(1)+1}\log\|B\|)$ bit operations.
