A multi-dimensional functional equation having cubic forms as solutions

In this paper, we obtain some results on the mvariable cubic functional equation \[f(2x_1 + y_1,..., 2x_m + y_m) + f(2x_1 y_1,..., 2x_m y_m)\\= 2f(x_1 + y_1,..., x_m + y_m) + 2f(x_1 y_1,..., x_m y_m) + 12f(x_1,..., x_m).\]The cubic form \(f(x_1,..., x_m) =\sum_{1\leq i\leq j\leq k\leq m} a_{ijk}x_ix_jx_k\) is a solution of the above functional equation.