A random polynomial with multiplicative coefficients is almost surely irreducible

Péter P. Varjú; Max Wenqiang Xu

A random polynomial with multiplicative coefficients is almost surely irreducible

Number Theory 2025-11-07 v1 Combinatorics Probability

Authors: Péter P. Varjú , Max Wenqiang Xu

Abstract

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability $1-O(d^{-1/2+\varepsilon})$ .

Keywords

polynomial theory polynomials over finite fields orthogonal polynomials

Cite

@article{arxiv.2511.04240,
  title  = {A random polynomial with multiplicative coefficients is almost surely irreducible},
  author = {Péter P. Varjú and Max Wenqiang Xu},
  journal= {arXiv preprint arXiv:2511.04240},
  year   = {2025}
}

Comments

19 pages

Related papers

View all related →