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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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