对于任意域k和任意整数m，n，其中0 <= 2 m <= n+1，设W_n是序列（x_0，...，x_n），设H_m是W_n中满足m次线性递归的序列的子集，即存在a_0，.，k中的a_m，不全为零，使得sum（a_i x_{i+j}，i=0.. m）= 0对于每个j= 0，1，.，嗯。等价地，H_m是（x_0，...，x_n），使得具有（i，j）项x_{i+j}（i=0. m，j=0. n-m）具有至多m的秩。我们用初等线性代数和多项式代数来研究这些集合H_m。特别地，当k是q元有限域时，我们将H_m的特征函数记为W_n中m维和m+1维线性子空间的特征函数的线性组合。我们推导了该特征函数的离散傅里叶变换（DFT）公式，并得到了一些结论。例如，如果m+1阶汉克尔方阵的2 m +1个元素是独立于k上的固定但不一定均匀的分布μ选择的，则当m->infty时，如果μ的DFT具有小于sqrt（q）的l_1范数，则矩阵是奇异的，概率接近1/q。如果q是一个正方形，那么这个界sqrt（q）是最好的。
英文摘要：
For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which there exist a_0,...,a_m in k, not all zero, such that sum(a_i x_{i+j}, i=0..m) = 0 holds for each j=0,1,...,n-m. Equivalently, H_m is the set of (x_0,...,x_n) such that the (m+1)-by-(n-m+1) matrix with (i,j) entry x_{i+j} (i=0..m, j=0..n-m) has rank at most m. We use elementary linear and polynomial algebra to study these sets H_m. In particular, when k is a finite field of q elements, we write the characteristic function of H_m as a linear combination of characteristic functions of linear subspaces of dimensions m and m+1 in W_n. We deduce a formula for the discrete Fourier transform (DFT) of this characteristic function, and obtain some consequences. For instance, if the 2m+1 entries of a square Hankel matrix of order m+1 are chosen independently from a fixed but not necessarily uniform distribution mu on k, then as m->infty the matrix is singular with probability approaching 1/q provided the DFT of mu has l_1 norm less than sqrt(q). This bound sqrt(q) is best possible if q is a square.
Satisfying sequences for rainbow partite matchings
Moscow Institute of Physics and Technology, St. Petersburg State University, Innopolis University, Russia（莫斯科物理技术学院，圣彼得堡国立大学，因诺波利斯大学）；Weizmann Institute of Science, Rehovot, Israel（魏茨曼科学研究所）
