Irreducibility of Binomials

Year 2023, , 62 - 70, 10.07.2023

Haohao Wang Jerzy Wojdylo Peter Oman

https://doi.org/10.24330/ieja.1260484

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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