Irreducibility of random polynomials: general measures

TL;DR

This paper demonstrates that random polynomials with coefficients from certain measures are almost surely irreducible and have Galois groups of full symmetric or alternating type as degree grows, under various conditions.

Contribution

It establishes irreducibility and Galois group properties of random polynomials with coefficients from finite-support measures, extending previous results to broader coefficient distributions.

Findings

Polynomials have no small-degree divisors with high probability.

Random polynomials are irreducible with probability tending to 1 under certain measures.

Galois groups are typically symmetric or alternating for large degree.

Abstract

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to $\mu$ while ensuring that $f(0)\neq0$, then there is a positive constant $\theta=\theta(\mu)$ such that $f(x)$ has no divisors of degree $\le \theta n$ with probability that tends to 1 as $n\to\infty$. Furthermore, in certain cases, we show that a random polynomial $f(x)$ with $f(0)\neq0$ is irreducible with probability tending to 1 as $n\to\infty$. In particular, this is the case if $\mu$ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of $[-H,H]\cap\mathbb{Z}$ of cardinality $\ge H^{4/5}(\log H)^2$ with $H$ sufficiently large. In addition, in all of these settings, we show that the Galois group of $f(x)$ is either $\mathcal{A}_n$ or $\mathcal{S}_n$ with high probability. Finally, when $\mu$ is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial $f(x)$ as above is irreducible with probability $\ge\delta$ for some constant $\delta=\delta(\mu)>0$. In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of $f(x)$ is $\mathcal{A}_n$ or $\mathcal{S}_n$ with probability $\ge\delta$.