Irreducibility of Littlewood polynomials of special degrees

TL;DR

This paper proves unconditionally that the probability of a random Littlewood polynomial being irreducible approaches 1 as the degree increases, confirming a long-standing folklore conjecture without relying on GRH.

Contribution

It establishes the unconditioned limit superior of the irreducibility probability of Littlewood polynomials as the degree tends to infinity.

Findings

Probability of irreducibility approaches 1 for large degrees

Confirms folklore conjecture unconditionally

Advances understanding of polynomial irreducibility in random settings

Abstract

Let $f$ be sampled uniformly at random from the set of degree $n$ polynomials whose coefficients lie in $\{ \pm 1\}$. A folklore conjecture, known to hold under GRH, states that the probability that $f$ is irreducible tends to $1$ as $n$ goes to infinity. We prove unconditionally that $$\limsup_{n \to \infty} \mathbb{P}(f \text{ is irreducible}) = 1.$$