A Functional Equation Originating from Elliptic Curves

Abstract

We obtain the general solution and the stability of the functional equation $f\left(x+y+z,u+v+w\right)+f\left(x+y-z,u+v+w\right)+2f\left(x,u-w\right)+2f\left(y,v-w\right)=f\left(x+y,u+w\right)+f\left(x+y,v+w\right)+f\left(x+z,u+w\right)+f\left(x-z,u+v-w\right)+f\left(y+z,v+w\right)+f\left(y-z,u+v-w\right)$. The function $f\left(x,y\right)=x^3+ax+b-y^2$ having level curves as elliptic curves is a solution of the above functional equation.