The fastest known algorithms for dealing with structured matrices, in the sense of the displacement rank measure, are randomized. For handling classical displacement structures, they achieve the complexity bounds $\tilde{O}(α^{ω-1} n)$ for solving linear systems and $\tilde{O}(α^2 n)$ for computing the nullspace. Here $n \times n$ is the size of the square matrix, $α$ is its displacement rank, $ω> 2$ is a feasible exponent for matrix multiplication, and the notation $\tilde{O}(\cdot)$ counts arithmetic operations in the base field while hiding logarithmic factors. These algorithms rely on an adaptation of Strassen's divide and conquer Gaussian elimination to the context of structured matrices. This approach requires the input matrix to have generic rank profile; this constraint is lifted via pre- and post-multiplications by special matrices generated from random coefficients chosen in a sufficiently large subset of the base field. This work introduces a fast and deterministic approach, which solves both problems within $\tilde{O}(α^{ω-1} (m+n))$ operations in the base field for an arbitrary rectangular $m \times n$ input matrix. We provide explicit algorithms that instantiate this approach for Toeplitz-like, Vandermonde-like, and Cauchy-like structures. The starting point of the approach is to reformulate a structured linear system as a modular equation on univariate polynomials. Then, a description of all solutions to this equation is found in three steps, all using fast and deterministic operations on polynomial matrices. Specifically, one first computes a basis of solutions to a vector M-Padé approximation problem; then one performs linear system solving over the polynomials to isolate away unwanted unknowns and restrict to those that are actually sought; and finally the latter are found by simultaneous M-Padé approximation.

翻译：针对位移秩意义下的结构矩阵，目前已知最快的算法均为随机算法。处理经典位移结构时，求解线性系统的复杂度为 $\tilde{O}(α^{ω-1} n)$，计算零空间的复杂度为 $\tilde{O}(α^2 n)$。其中 $n \times n$ 为方阵尺寸，$α$ 为位移秩，$ω> 2$ 为矩阵乘法的可行指数，记号 $\tilde{O}(\cdot)$ 表示基域中算术运算次数并忽略对数因子。这些算法基于 Strassen 分治高斯消元法在结构矩阵场景下的适应性改造，要求输入矩阵具有一般秩轮廓；该约束通过预乘和后乘由基域足够大子集中随机系数生成的特殊矩阵予以解除。本文提出一种快速确定性方法，对任意 $m \times n$ 矩形输入矩阵，可在基域 $\tilde{O}(α^{ω-1} (m+n))$ 次运算内同时解决上述两个问题。我们给出了针对 Toeplitz 型、Vandermonde 型与 Cauchy 型结构的具体实现算法。该方法的起点是将结构化线性系统转化为单变量多项式上的模方程，随后通过三步快速确定性操作（均基于多项式矩阵）求得该方程的所有解描述：首先计算向量 M-Padé 逼近问题的解基，然后通过多项式域上的线性系统求解分离无关未知量、限制至待求变量，最后通过联立 M-Padé 逼近获得目标解。