Distribution of reducible polynomials with a given coefficient set

Abstract

For a given set of integers $\mathcal{S}$, let ${\mathcal{R}}_n^{\ast }(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_n{X}^n+a_{n-1}{X}^{n-1}+\cdots +a_{1}X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in \mathcal{S}$ and $a_0{a}_n\ne 0$. In this note, we shall give an explicit bound of $|{\mathcal{R}}_n^{\ast }(\mathcal{S})|$. We also present an application of this bound to reducible bivariate polynomials over $\mathbb{Z}[X,Y]$.