Probabilistic Galois Theory: The square discriminant case
L Bary‐Soroker, O Ben‐Porath… - Bulletin of the London …, 2024 - Wiley Online Library
L Bary‐Soroker, O Ben‐Porath, V Matei
Bulletin of the London Mathematical Society, 2024•Wiley Online LibraryThe paper studies the probability for a Galois group of a random polynomial to be A n 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 {-L,...,L}. The state‐of‐the‐art upper
bound is O (L− 1) O(L^-1), due to Bhargava. We conjecture a much stronger upper bound L−
n/2+ ε 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.
We focus on the so‐called large box model, where we choose the coefficients of the
polynomial independently and uniformly from− L,…, L {-L,...,L}. The state‐of‐the‐art upper
bound is O (L− 1) O(L^-1), due to Bhargava. We conjecture a much stronger upper bound L−
n/2+ ε 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.
Abstract
The paper studies the probability for a Galois group of a random polynomial to be An$A_n$. We focus on the so‐called large box model, where we choose the coefficients of the polynomial independently and uniformly from {−L,…,L}$\lbrace -L,\ldots , L\rbrace$. The state‐of‐the‐art upper bound is O(L−1)$O(L^{-1})$, due to Bhargava. We conjecture a much stronger upper bound L−n/2+ε$L^{-n/2 +\epsilon }$, 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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