Consider $n$ real vectors of arbitrary dimension $x_i$ with $i\in \{1,2,...,n\}$. Furthermore, consider a function $f(x_1,x_2,x_3)$ where actually the function can only depend on scalars $x_i\cdot x_j$ so that $i,j\in \{1,2,3\}$ count as input variables and other $x_i$ count as parameters. I am interested to find out what kinds of functions $f(x_1,x_2,x_3)$ sum to zero under cyclic index permutations:
$$0=f(x_1,x_2,x_3)+f(x_2,x_3,x_1)+f(x_3,x_1,x_2)$$
The last two arguments in each case may be permutation invariant:
$$f(x_1,x_2,x_3)=f(x_1,x_3,x_2)$$
etc. Does anyone know of a class of functions that have this behavior? Or maybe there is a procedure that can be applied to figure this out? Thanks for any suggestion.