Probabilistic Galois Theory -- The Square Discriminant Case

TL;DR

This paper investigates the probability that a random polynomial's Galois group is the alternating group, focusing on the square discriminant case, and proposes conjectures and bounds for these probabilities.

Contribution

It introduces a conjecture for a sharper upper bound on the probability and provides strong lower bounds for the probability of the Galois group being $A_n$ and the discriminant being a square.

Findings

Proposes a conjecture for an upper bound of $L^{-n/2 + \\epsilon}$.

Establishes strong lower bounds on the probability of the Galois group being $A_n$.

Analyzes the probability of the discriminant being a square in the large box model.

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 $\{-L,\ldots, L\}$. The state-of-the-art upper bound is $O(L^{-1})$, due to Bhargava. We conjecture a much stronger upper bound $L^{-n/2 +\epsilon}$, 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.