Irreducibility of Binomials

Abstract

In this paper, we prove that the family of binomials $x_1^{a_1}
 \cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1,
 \ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying
 the connection between the irreducibility of a binomial in
 ${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb
 C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that
 the necessary and sufficient conditions for the irreducibility of
 this family of binomials is equivalent to the existence of a
 unimodular matrix $U_i$ with integer entries such that $(a_1,
 \ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots,
 m+n\}$, where $\be_i$ is the standard basis vector.