Shane Chern : Distribution of Reducible Polynomials with a Given Coefficient Set, p. 141-146


For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this note, we shall give an explicit bound of $\vert\mathcal{R}_n^*(\mathcal{S})\vert$. We also present an application of this bound to reducible bivariate polynomials over $\mathbb{Z}[X,Y]$.

Key Words: Reducible polynomial, bivariate polynomial, counting function, Euler's identity

2010 Mathematics Subject Classification: Primary 11C08
Secondary 11N45