Polynomial Equations and Rank of Matrices Over F_2 Related to Persymmetric Matrices

Exponential sums and rank of double persymmetric matrices over F_2

We obtain, using exponential quadratic sums, explicit expressions for the number of double persymmetric matrices with entries in F_2 of given rank. (A matix [a(i,j)) is persymmetric if a(i,j) = a(r,s) for i+j = r+s)

On a conjecture concerning enumeration of 2n x k n-times persymmetric matrices over F_2 by rank

In this paper we announce a conjecture concerning enumeration of 2n x k n-times persymmetric matrices over F_2 by rank.

On a conjecture concerning the fraction of invertible m-times persymmetric matrices over F_2

In this paper we state that the fraction of invertible m-times persymmetric matrices over F_2 is equal to \prod_{j=1}^{m}(1-2^{-j})

Exponential sums and rank of persymmetric matrices over F_2

Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s).

Enumeration of some particular 2N x 9 N-times persymmetric matrices overt F_2 by rank

In this paper we count the number of some particular 2n x 9 n-times rank i matrices over F_2.

Enumeration of some particular 2Nx10 N-times persymmetric matrices over F_2 by rank

In this paper we count the number of some particular 2nx10 n-times rank i matrices over F_2.

Rank of matrices with entries from a multiplicative group

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.

有限秩乘法群上的矩阵计数

Counting matrices over finite rank multiplicative groups

该研究针对特征零域有限秩乘法群有限子集上的矩阵，推导了给定秩、行列式及特征多项式的矩阵数量的上界，其秩相关结果可视为Alon和Solymosi工作的统计版本。

有理矩阵的秩、行列式和特征多项式的统计学

Statistics of ranks, determinants and characteristic polynomials of rational matrices

研究有理矩阵的秩、行列式和特征多项式的统计特性，通过建立上界和下界来计数具有特定属性的矩阵数量。

Parallel computation of the rank of large sparse matrices from algebraic K-theory

This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z…