I'm trying to represent the following in the elementary symmetric functions base:
$ \sum\limits_{i \neq j} x^2_i x_j $
and
$ \sum\limits_{i \neq j} x_i^2 x_j^2 $
I don't really know how to algorithmically approach the problem. All I've managed so far was doing some guessing, but it has done no good
As a hint, the first has degree $3$. So you need a linear combination of things of degree $3$. Note also that there are no terms involving cubes.
A degree three symmetric problem can always be solved by reference to the same problem with just three variables (this picks out the relevant terms, and if you go to four or more variables you only get terms of the same type, and the coefficients work out right).
The forms you have to work with are $(x+y+z)^3$, $(x+y+z)(xy+xz+yz)$ and $xyz$. The first has cubic terms which can't be cancelled by the other forms. From there it is pretty easy.
For the second one, of degree $4$, it is more complex - you need four variables, and there are more possibilities. But note that there are no fourth powers, and no cubes, which reduces the possibilities once again to manageable proportions.