On the representation of integers by binary forms defined by means of the relation (x + yi)n= Rn(x,y) + Jn(x,y)i

Abstract

Let F be a binary form with integer coefficients, degree d ≥ 3 and nonzero discriminant. Let RF (Z) denote the number of integers of absolute value at most Z which are represented by F . In 2019 Stewart and Xiao proved that RF (Z) ∼ CFZ 2/d for some positive number CF . We compute CRn and CJn for the binary forms Rn(x, y) and Jn(x, y) defined by means of the relation (x+ yi) = Rn(x, y) + Jn(x, y)i, where the variables x and y are real.