Probabilistic Galois Theory: The square discriminant case

The paper studies the probability for a Galois group of a random polynomial to be (Formula presented.). We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from (Formula presented.). The state-of-the-art upper bound is (Formula presented.), due to Bhargava. We conjecture a much stronger upper bound (Formula presented.), 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.