概率伽罗瓦理论：平方判别情况

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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摘要

本文研究了随机多项式的伽罗瓦群是 .我们将重点放在所谓的大箱模型上，在这个模型中，我们独立地、均匀地从 . 中选择多项式的系数，最先进的上界是由 Bhargava 提出的 .我们猜想有一个更强的上界，而且这个上界本质上是尖锐的。我们证明了这个概率以及相关的判别式为平方的概率的强下界。

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Probabilistic Galois Theory: The square discriminant case

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.