On the Representation of Integers by Binary Forms Defined by Means of the Relation $(x + yi)^n = R_n(x, y) + J_n(x, y)i$

Abstract

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d \geq 3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. In 2019 Stewart and Xiao proved that $R_F(Z) \sim C_FZ^{2/d}$ for some positive number $C_F$. We compute $C_{R_n}$ and $C_{J_n}$ for the binary forms $R_n(x, y)$ and $J_n(x, y)$ defined by means of the relation $ (x + yi)^n = R_n(x, y) + J_n(x, y)i$, where the variables $x$ and $y$ are real.