4 problems

For each $n$, let $\Sigma_n=(\sigma_{n,1},\ldots,\sigma_{n,n})$, where each $\sigma_{n,j}$ is a subset of $[n]=\{1,2,\ldots,n\}$. Let $X_n\in\operatorname{M}_n(\mathbb{Z}_p)$ be Ha…

Let $X_m:=\{x_1,\ldots,x_m\}$ be a finite ordered subset of $\mathbb{Z}_p$ whose elements have distinct reductions modulo $p$, and let…

Let $P_{1}(t),\dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ be monic polynomials whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. Let $q_{j}=p^{\deg(P_{j})}$, let…

Let $P_{1}(t),dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ be monic polynomials whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. For each $j$, let $G_{j}$ be a fi…