On a functional equation arising from number theory

This work aims to determine the general solution f : F 2 → K of the functional equation f ( ϕ ( x , y , u , v ) ) = f ( x , y ) f ( u , v ) for suitable conditions on the function ϕ : F 4 → F 2 , where F will denote either R or C , and K is an abelian group. Using this result, we determine the solution f : C 2 → C ⋆ of the functional equation f ( u x − v y , u y + v ( x + y ) ) = f ( x , y ) f ( u , v ) for all x , y , u , v ∈ C without assuming any regularity condition. Here ( C ⋆ , ⋅ ) is the multiplicative group of nonzero complex numbers.