论文标题
概率的Galois理论 - 正方形判别案例
Probabilistic Galois Theory -- The Square Discriminant Case
论文作者
论文摘要
该论文研究了一个随机多项式的Galois组为$ a_n $。我们专注于所谓的大型框模型，在其中我们从$ \ { - l，\ ldots，l \} $独立和均匀地选择多项式的系数。由于Bhargava，最新的上限为$ O（l^{ - 1}）$。我们猜测上限$ l^{ - n/2 +ε} $要强得多，并且该界限本质上是锋利的。我们证明，在这种概率和判别剂是正方形的相关概率上都具有强大的下限。
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 $L^{-n/2 +ε}$, 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.
