Probabilistic Galois Theory -- The Square Discriminant Case

Abstract

The paper studies the probability for a Galois group of a random polynomial to be $A_n$. We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from $\{-L,\dots ,L\}$. The state-of-the-art upper bound is $O(L^{-1})$, due to Bhargava. We conjecture a much stronger upper bound $L^{-n/2+\hat{I}\mu }$, and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square.