My research field is combinatorics. I am not very good at Algebra. So I want to ask for a given real multivariate polynomial $f(x_1,x_2,\cdots,x_n)$, is there any algebraic method to compute whether the following sum
$$\sum_{(x_1,x_2,\cdots,x_n)\in \{0,1\}^n}f(x_1,x_2,\cdots,x_n)$$
is nonzero?
Let $c_{\alpha}$ be the coefficient of $x^\alpha$ (using multi-index notation). The sum of $x^{\alpha}$ over $x \in \{0,1\}^n$ is $2^{Z(\alpha)}$ where $Z(\alpha)$ is the number of entries $\alpha_i = 0$. So you want to compute $\sum_\alpha c_\alpha 2^{Z(\alpha)}$.
Since $0^2 = 0,$ and $1^2=1,$ for every monomial you compute its support (all the variables appearing to a non-zero power), and group together the monomials with the same support, to get a new polynomials, where every variable has degree $0$ or $1.$ Now you just have a function defined on the power set $P_n$ of $\{1, \dotsc, n\}$ and you are asking whether its sum over $P_n$ is $0.$ I don't see anything easier or faster than just summing it (the fact that $P_n$ is the power set of something seems irrelevant).
