2 problems

Let $n$ be a positive integer. For $0\leq i\leq \frac{n+1}{3}$ and $0\leq j\leq \frac{n+1}{3}$, let $\widetilde{\mathcal{P}}_{i,j}(n)$ be the matrix obtained by stacking the inclus…

Unimodular density conjecture. The probability that a uniformly random element of $M_{n,k}({\mathbb F}_q[x];m)$ is unimodular is given by