Reduction of $f:\mathbb{C}^n\to \mathbb{C}$ into sum of $f_{ij}:\mathbb{C}^2\to\mathbb{C}$ I was browsing wikipedia the other day when I came across the following (paraphrased) claim:
$$
\exists f_{ij}:\mathbb{C}^2\to \mathbb{C} \mbox{ s.t. }
 f(x_1,\dots,x_n)=\sum_{i,j} f_{ij}(x_i,x_j)
$$
Now for the life of me, I can't find that page and I'm not sure how I would search the literature for such a claim. I'm having trouble believing it unconditionally, but I would like to know under which conditions it is true.
 A: For $n=3$, the existence of the desired decomposition of $f$ is equivalent to the equation
$$
f(x_1,x_2,x_3)+f(x_1,x_2',x_3')+f(x_1',x_2,x_3')+f(x_1',x_2',x_3)\\
=f(x_1,x_2,x_3')+f(x_1,x_2',z_3)+f(x_1',x_2,x_3)+f(x_1',x_2',x_3')
$$
holding identically.
It can be directly checked that, if $f$ has a decomposition of the desired form, then the identity above holds.  Conversely, if the identity above holds, then (setting $x_1'=x_2'=x_3'=0$):
$$
f(x_1,x_2,x_3)=f(x_1,x_2,0)+f(x_1,0,x_3)+f(0,x_2,x_3)-f(x_1,0,0)-f(0,x_2,0)-f(0,0,x_3)+f(0,0,0)
$$
where we have expressed $f(x_1,x_2,x_3)$ as a sum of functions, each of which depends on at most two of the variables.
I would guess there are similar identities (or families of identities) for $n\geq 4$, but I'm not sure what they are.
A: This is an ad hoc expansion of Pink Elephants' method, if anyone's still paying attention to this question.
Given $f: \mathbb{C}^n \to \mathbb{C}$, define the "Euler characteristic" as the function $\chi(f)$ in $2n$ variables given by
$$
\chi(f) = f(x_1, \ldots, x_n) - \sum_i f(x_1, \ldots, x_i',x_{i+1}, \ldots, x_n) + \sum_{i,j} f(x_1, \ldots, x_i', \ldots, x_j', \ldots, x_n)  - \ldots + (-1)^nf(x_1', x_2', \ldots, x_n').
$$
Then it looks like $f$ is a sum of functions from $\mathbb{C}^{n-1}$ if and only if $\chi(f) = 0$ (identically). You could iterate the definition of $\chi$ to get a criterion to reduce from $n$ to $2$. 
