Galois groups of random polynomials of large degree

Source abstract

We study random polynomials of the form $R(x)=x^n+ω_{n-1}x^{n-1}+\cdots+ω_0$, where $ω_0,\dots,ω_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $ω_1,\dots,ω_{n-1}$ have a fixed common law $μ$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies $H_2(μ)=-\log\|μ\|_2^2>12$, then $\mathbb{P}(\operatorname{disc}(R)\text{ is a square})=O_μ(1/\log n)$. Combined with previous results, this shows that, for such measures $μ$ and under additional hypotheses, $R$ has full Galois group $\mathrm{Sym}(n)$ with high probability, when conditioned on $ω_0\ne 0$.