6 problems

Frost–Storey conjecture. The matrix $A$ is equivalent to its Smith normal form if and only if the reduced minors of every order generate the unit ideal in the polynomial ring.

Genericity conjecture. For $m\geq 3$, $\mathcal{D}_n^{\operatorname{gen}(m)}$ consists exactly of the $m\times(n+1)$ $\{0,1\}$-matrices such that, for every $k\in[0,n]$, there are…

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

Elementary-divisor transformation conjecture. Then $(y-y_0)^{m\cdot\ell}$ is an elementary divisor of $Q(y)$.

For each $n$, consider the $(n+1)\times n$ matrix of the form … with the evident $(n+1)\times n$ generalization, where the exponents are positive integers subject to the correspond…

Polynomial-matrix rank conjecture. The matrix $A_{a_1,a_2,\dots,a_n}(z)$ should have rank smaller than $n-1$ if and only if $z^f=1$. This generalizes the proved three-exponent case…