Galois groups of random integer matrices

Theresa C. Anderson; Evan M. O'Dorney

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arXiv · 2506.06463

Galois groups of random integer matrices

Theresa C. Anderson·Evan M. O'Dorney

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Abstract

We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows $M_n(T) \gg T^{n^2 - n + 1} \log T$, which we conjecture is sharp. We first use the large sieve to get $M_n(T) \ll T^{n^2 - 1/2}\log T$. Using Fourier analysis and the geometric sieve, as in Bhargava's proof of van der Waerden's conjecture, we improve this bound for some classes of $A$.

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Theresa C. Anderson, Evan M. O'Dorney. 2025-06-06. Galois groups of random integer matrices. https://arxiv.org/abs/2506.06463

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