Let $P(x_1,x_2,\ldots,x_n)$ be a multilinear polynomial of $n$ (real or complex) variables.
As I see, it can be represented in the form $$ P(x_1,x_2,\ldots,x_n)=\sum_{(\alpha_1, \alpha_2, \ldots \alpha_n)\in \{0,1\}^{n}}C_{\alpha_1, \alpha_2, \ldots, \alpha_n} x_{1}^{\alpha_1} x_{2}^{\alpha_2} \ldots x_{n}^{\alpha_n}, \tag{1} $$ or in the form $$ P(x_1,x_2,\ldots,x_n)=c+\sum_{1 \leq i_1 < i_2 <\ldots< i_k \leq n \atop k=\overline{1,n}}C_{i_1, i_2, \ldots, i_k} x_{i_1} x_{i_2} \ldots x_{i_k}. \tag{2} $$ Of course, in (1) we assume, by definition, $0^0=1$. It is obviously that sums in (1) and (2) consist of $2^n$ terms.
How multilinear polynomials can be characterized? Are there criteria for a polynomial to be multilinear?