Probabilistic Galois Theory -- The Square Discriminant Case

2022/07/25 by Bary-Soroker, Lior, Ben-Porath, Or, Matei, Vlad · 2 citations
#11R32 #12F12 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.12493

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,…, 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 +ε}, 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.

Cited by

• Irreducibility and Galois groups of random reciprocal polynomials of large degree
• Law of large numbers for the discriminant of random polynomials

Related

• Probabilistic Galois Theory in Function Fields
• Galois Groups of Generic Polynomials
• Galois groups of random polynomials over the rational function field
• Full Galois groups of polynomials with slowly growing coefficients
• Galois groups of random integer matrices