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 Rényi entropy of order $2$ satisfies $H_2(\mu )=-\log \Vert \mu {\Vert }_2^2>12$, then $\mathbb{P}(\mathrm{disc}(R)\text{ is a square})=O_{\mu }(1/\log n)$. 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 ${\omega }_0\ne 0$.

39 pages, comments welcome!