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In this paper, we obtain some results on the m-variable cubic functional equation f(2x1 + y1, · · · , 2xm + ym) + f(2x1 − y1, · · · , 2xm − ym) = 2f(x1 + y1, · · · , xm + ym) + 2f(x1 − y1, · · · , xm − ym) + 12f(x1, · · · , xm). The cubic form f(x1, · · · , xm) = ∑ 1≤i≤j≤k≤m aijkxixjxk is a solution of the above functional equation. c ©2016 all rights reserved.

A multi-dimensional functional equation having cubic forms as solutions