Probabilistic Galois Theory: The square discriminant case

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-04-24 DOI:10.1112/blms.13049

Lior Bary-Soroker, Or Ben-Porath, Vlad Matei

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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.