Babai–Vu–Wood irreducibility conjecture for random characteristic polynomials

Let $\mu$ be a fixed nontrivial finitely supported measure on $\mathbf{Z}$, let $M$ be an $n\times n$ random matrix whose entries are independent with distribution $\mu$, and let

$$\varphi(t)=\det(t-M)$$

be its characteristic polynomial. Here, “with high probability” means with probability $1-o(1)$ as $n\to\infty$.

Babai–Vu–Wood conjecture. $\varphi$ is irreducible with high probability.

This conjecture concerns the typical algebraic structure of characteristic polynomials of discrete random matrices. The paper proves it under certain conditions, including a four-prime uniformity hypothesis and, conditionally, the extended Riemann hypothesis; the unrestricted statement remains open in the source.