Irreducibility of lacunary polynomials with 0,1 coefficients

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 $F(x)=1+{\sum }_{i=1}^k{x}^{n_i}$, where $0<n_1<\cdots <n_k\le N.$ 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.