Galois groups of random polynomials of large degree

Abstract

We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the R\'{e}nyi entropy of order $2$ satisfies $H_2(\mu)=-\log\|\mu\|_2^2>12$, then $\mathbb{P}(\operatorname{disc}(R)\text{ is a square})=O_\mu(1/\log n)$. Combined with previous results, this shows that, for such measures $\mu$ and under additional hypotheses, $R$ has full Galois group $\mathrm{Sym}(n)$ with high probability, when conditioned on $\omega_0\ne 0$.