Irreducibility of Binomials

Authors : Haohao WANG, Jerzy WOJDYLO, Peter OMAN

Pages : 62-70

Doi:10.24330/ieja.1260484

View : 47 | Download : 37

Publication Date : 2023-07-10

Article Type : Research Paper

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 $\\gcdinsert ignore into journalissuearticles values(a_1, \\ldots, a_m, b_1, \\ldots, b_n);=1$ 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 ${\\mathbb C}insert ignore into journalissuearticles values(x_2, \\ldots, x_m, y_1, \\ldots, y_n);[x_1]$. 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 $insert ignore into journalissuearticles values(a_1, \\ldots, a_m, b_1, \\ldots, b_n);^T=U_i \\be_i$ 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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