Irreducibility of lacunary polynomials with 0,1 coefficients

TL;DR

This paper proves that high-degree, sparse polynomials with coefficients 0 or 1 are almost surely irreducible as their degree and number of terms grow large, highlighting their typical algebraic complexity.

Contribution

It establishes a probabilistic irreducibility result for lacunary polynomials with 0,1 coefficients, extending understanding of their algebraic structure in the limit.

Findings

High probability of irreducibility for large degree and few terms

Asymptotic irreducibility as degree and number of terms grow

Probabilistic model based on uniform count of such polynomials

Abstract

We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that $\lim_{k\rightarrow\infty}\limsup_{N\rightarrow\infty}\mathbb{P}(\text{$F(x)$ is reducible})=0.$ The probability in this context is derived from the uniform count of polynomials $F(x)$ of the above form.