Irreducibility of Littlewood polynomials of special degrees

TL;DR

The paper proves an unconditional limsup result for the irreducibility of random Littlewood polynomials along subsequences with degree $n=p^r-1$. It combines a $2$-adic Newton polygon approach for $p=2$ with a Bombieri–Koukoulopoulos–Kouksoulopoulos (BKK) framework to rule out small-degree factors, and extends the argument to large primes $p>1470$ via cyclotomic factorization modulo $2$ together with a numerical verification to satisfy BKK conditions. For the intermediate prime range $7\le p\le 1470$, it develops an approximate equidistribution analysis modulo $3$ and a refined divisors-control mechanism to bound the probability of any low-degree divisor, again leveraging the BKK machinery. Altogether, the paper establishes that $\limsup_{n\to\infty} \mathbb{P}(f \text{ irreducible})=1$ along an explicit infinite subsequence, advancing unconditional understanding beyond GRH-conditional results.

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 $$\limsup_{n \to \infty} \mathbb{P}(f \text{ is irreducible}) = 1.$$

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• proof : Proof of Theorem \ref{['thm:main']} for $p > 1470$.
• Lemma 4.1
• proof
• Definition 4.2
• Lemma 4.3
• proof