Irreducibility of polynomials with random multiplicative coefficients revisited

TL;DR

This paper proves that as the degree of a polynomial with random multiplicative coefficients in \\{ extpm 1\\} increases, the probability that the polynomial is irreducible approaches 1, highlighting a typical irreducibility behavior.

Contribution

It establishes the asymptotic irreducibility of polynomials with random completely multiplicative \\{ extpm 1\\} coefficients, revisiting and strengthening previous results.

Findings

Probability of irreducibility tends to 1 as degree increases

Random multiplicative coefficients lead to high irreducibility likelihood

Supports conjecture on typical irreducibility of such polynomials

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 \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is irreducible}\big] = 1. \]