Irreducibility of polynomials with random multiplicative coefficients revisited

TL;DR

The paper investigates irreducibility of polynomials with dependent, multiplicative-coefficient randomness by studying $P_{f,N}(X)=\sum_{n=1}^N f(n) X^{n-1}$ where $f$ is a random completely multiplicative function taking values in $\{-1,1\}$. The authors prove an unconditional result along the subsequence $N=2^k$: as $k\to\infty$, $P_{f,N}(X)$ is irreducible with probability tending to 1, using a shift $\tilde{P}_{f,N}(X)=P_{f,N}(X+1)$, parity-control via a special index set and Lucas's theorem, a Newton polygon argument over $\mathbb{Q}_2$, and bounds on potential low-degree factors (via a Rourke-type result and Berry–Esseen concentration). They also derive an unconditional corollary for Turyn polynomials $F_{d,p}(z)=\sum_{a=1}^d (\frac{a}{p}) z^a$, showing irreducibility with high probability as $k\to\infty$ when $d=2^k$ and $p$ ranges over suitably large primes. The work advances understanding of irreducibility in dependent-random coefficient models and connects to classical objects in number theory, illustrating universal behavior beyond independent-coefficient settings. The results rely on a blend of probabilistic, analytic, and algebraic tools, including prime distributions in progressions, Newton polygons, and Mahler-measure bounds. This contributes to the broader goal of characterizing irreducibility in random polynomial families with structured dependence.

Abstract

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is irreducible}\big] = 1. \]

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4