All Questions

My question is somewhat related to this question. Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...

Let the $N$ by $N$ matrix $A$ be defined by the tetration: $$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...

Let $A$ be a $2n$-by-$2n$ matrix with entries in $\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$ has rank $\...

I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix: $$M=\begin{bmatrix} x_1 &...

Given $a,m,n,t\in\Bbb Z$, with $n=m^t$ and $a$ arbitrary, and given $\mathbb{Z}$-linearly independent vectors $v_1,\dots,v_n\in\Bbb Z^n$, and an arbitrary vector $w\in\Bbb Z^n$, such that $$\langle ...

Let $n,H$ two fixed positive integers. Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...