Full Galois groups of polynomials with slowly growing coefficients

TL;DR

This paper proves that for random monic polynomials with integer coefficients growing slowly to infinity, the Galois group is almost surely the full symmetric group as degree increases.

Contribution

It establishes the asymptotic full symmetric Galois group for polynomials with slowly growing coefficient bounds, extending previous results to new growth regimes.

Findings

Galois group is full symmetric when coefficient bounds grow slowly

Results extend to more general independent coefficients

The method applies for various growth rates of coefficient bounds

Abstract

Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group of $f$ is the full symmetric group, asymptotically almost surely, as $n\to \infty$. When $L$ grows rapidly to infinity, say $L>n^7$, this theorem follows from a result of Gallagher. When $L$ is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if $L< 17$, it is conditional on the general Riemann hypothesis). Hence the most interesting case of the theorem is when $L$ grows slowly to infinity. Our method works for more general independent coefficients.