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arXiv · 2207.12493

Probabilistic Galois Theory -- The Square Discriminant Case

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 $\{-L,\ldots, L\}$. 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.

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Lior Bary-Soroker, Or Ben-Porath, Vlad Matei. 2022-07-25. Probabilistic Galois Theory -- The Square Discriminant Case. https://arxiv.org/abs/2207.12493

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