On the irreducibility of the non-cyclotomic part of most 0,1-polynomials with few terms

2025/08/17 by Michael Filaseta, Filaseta, Michael, Alexandros Kalogirou +1
Computer Science · Mathematics · #11R09 (Primary) 11C08 #12E05 (Secondary) #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2508.12242

openalex publication_date 2025/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide an alternative exposition of a result due to Schinzel. Fix an integer k ≥ 2. For almost all choices of positive integers n₁ < ⋯ < n_k, we show that the polynomial F(x) = 1 + x^n₁ + ⋯ + x^n_k, removed of its cyclotomic factors, is irreducible.

Citations

• Irreducibility of lacunary polynomials with 0,1 coefficients
• A proof of the Schinzel-Zassenhaus conjecture on polynomials

Related

• A Note on Cyclotomic Polynomials
• An alternative proof for the irreducibility of the p-th cyclotomic polynomial
• Irreducibility of lacunary polynomials with 0,1 coefficients
• An irreducible class of polynomials over integers
• Cyclotomic polynomials without using the zeros of Yⁿ-1