Squarefree values of polynomial discriminants II

Abstract

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ``arithmetic Bertini theorem'' conjectured by Poonen for $\mathbb{P}^1_\mathbb{Z}$. Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree~$n$ having associated Galois group~$S_n$ and absolute discriminant less than $X$, improving the best previously known lower bound of $\gg X^{1/2+1/n}$. Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$-extensions of quadratic number fields of bounded discriminant.