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.

Keywords

• Multivariate polynomial
• irreducibility
• unimodular matrix

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