Let $$F(x_1,\dots,x_k) = \sum_{l_1 + \dots + l_k = n} C_{l_1,\dots,l_n}x_1^{l_1} x_2^{l_2} \dots x_k^{l_k}$$
Now all $x_i \in \{0,1\}$, i.e they are binary variables, I need to find or approximate all the coefficients $C_{l_1,\dots,l_n}$(these coefficients are known to be whole numbers) I can sample the value of $F(x_1,\dots,x_k)$ at various points, but finding its value at points where the majority of $x_i's$ are zero is much more computationally efficient than points where a large number of $x_i's $ are ones.
Can anyone point me in the direction to solving it? I am clueless about the literature on methods that exist for such approximations.
Also, feel free to reply if you know a method to approximate such functions but using a different functional form.