All Questions

Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the ...

By rank I imply rank over reals ($\mathbb R$). I consider two $n\times n$ matrices $A,B$ having entries in $0/1$. The product below follows usual matrix product rules except $xy$ is $AND(x,y)$ and $x+...

Let $\mathcal T_n=\{M\in\{0,1\}^{n\times n}:\mathsf{Per}(M)=\mathsf{Det}(M)\wedge\mathsf{Det}(M)\in\{0,1\}\}$ (restricted set unimodular or singular having permanent and determinant identical). $\...

Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...

All, Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is, \begin{equation*} \pi_{A}(x)=per(xI-A). \...

We know that the order of $SL_n({\mathbb F}_p)$ is $$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$ Dividing by $p^{n^2}$, we deduce the probability that $\det$ takes the value $1$ over $M_n({\mathbb ...