An asymptotic formula for Goldbach's conjecture with monic polynomials in $\mathbb{Z}[\theta ][x]$

Abstract

In this paper, we consider $D=\mathbb{Z}[\theta ]$, where $$θ= \sqrt{-k} \,\,\,\, \mbox{if}\;\;\;-k\not\equiv 1 \;(\mbox{mod}\;4)\,\,\,\,\mbox{or}\,\,\,\, θ=\frac{\sqrt{-k}+1}{2} \,\,\,\, \mbox{if}\;\;\;-k\equiv 1 \;(\mbox{mod}\;4),$$ $k\ge 2$ is a squarefree integer, and we proved that the number $R(y)$ of representations of a monic polynomial $f(x)\in \mathbb{Z}[\theta ][x]$, of degree $d\ge 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\mathbb{Z}[\theta ][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic to $(4y{)}^{2d-2}$.