Reducibility of polynomials \(a_0(x)+a_1(x)y+a_2(x)y^2\) modulo \(p\)
If \(f(x)=a_0(x)+a_1(x)y+a_2(x)y^2\in\mathbb{Z}[x,y]\) is an absolutely irreducible polynomial of degree \(m\) in \(x\), then the reduction of \(f\) modulo a given prime \(p\) will also be absolutely irreducible if \(p>c_mH(f)^{e_m}\) where \(H(f)\) is the height of \(f\), \(c_m\) is a specified constant depending on \(m\), and \(e_m=2m\) for \(m\geq 4\). The \(e_m\) are shown to be the best possible exponents if a certain conjecture of Bouniakowsky is assumed to be true.
scientific article; zbMATH DE number 1292447