Let $S_1\subset Z_2^{n_1}$ and $S_2\subset Z_2^{n_2}$, $S=(S_2,S_1)\subset Z_2^{n_1+n_2}$, I'd like to restrict $S_1$ and $S_2$ s.t The following relations hold:

$\Sigma_{1 \leq i \leq n_1+n_2}(-1)^{x_i}=0,$

$\Sigma_{1 \leq i,j \leq n_1+n_2,i\neq j}(-1)^{x_i+x_j}=0,$

$‎\vdots$‎

$\Sigma_{1 \leq i_1,i_2,...,i_l \leq n_1+n_2}(-1)^{x_{i_1}+x_{i_2}...+x_{i_l}}=0, $ for some $1 \leq l \leq n_1+n_2$,

Does anyone have any idea or seen any references? I should say it's about Boolean functions?