## A functional equation originating from elliptic curves.(English)Zbl 1146.39037

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

### MSC:

 39B32 Functional equations for complex functions 14H52 Elliptic curves 11G05 Elliptic curves over global fields 39B82 Stability, separation, extension, and related topics for functional equations
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### References:

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