Irreducibility of Random Polynomials of ℤ[X]

Abstract In a recent paper, Bary-Soroker, Koukoulopoulos, and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_{j}$ drawn independently according to measures $\mu _{j},$ then $A$ is irreducible with probability tending to $1$ as $n\to \infty $ under a condition of near-uniformity of the $\mu _{j}$ modulo four primes (which notably happens when the $\mu _{j}$ are uniform over a segment of $\mathbb{Z}$ of length $N\ge 35.$) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound $$ \begin{align*} &\mathbb{P}(A \textrm{ irreducible}) = 1 -O(1/\sqrt n)\end{align*} $$ when we have near-uniformity modulo twelve primes, which notably happens when the $\mu _{j}$ are uniform over a segment of length $N\ge 10^{8}.$