It is easy to see that if a function $f(x_1,x_2,x_3)$ can be written in the form: $$ f(x_1,x_2,x_3) = g(x_1,x_2) - g(x_1,x_3) + g(x_2,x_3) $$
for some function $g$, then we have: $$ f(x_1,x_2,x_3) - f(x_1,x_2,x_4) + f(x_1,x_3,x_4) - f(x_2,x_3,x_4) =0. $$ Only if this last equation is satisfied there is the possibility that $f$ can be written in the form above.
Suppose instead that we look for a function $f$ in the form: $$ f(x_1,x_2,x_3) = g(x_1,x_2) + g(x_1,x_3) + g(x_2,x_3). $$
Can we find a necessary condition for $f$ in a similar way? Can it be done for symmetric sums in arbitrary dimensions?
Thank you.
(Please help me finding appropriate tags, feel free to edit!)