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The problem to find polynomials $f(x,y)$ such that $f(x,y) - f(y,x) = 0$ can be 'solved' by characterizing the solutions as all polynomials in $xy$ and $x+y$ (according to the fundamental theorem of symmetric functions).

Is there an analogous approach for finding all $f$ such that, for example, $yf(x,y) - xf(y,x) = 0$? The most general setting I'm thinking of is an equation of the form $\sum_\sigma a_\sigma f^\sigma = 0$ where a polynomial $f(x_1,\dots,x_n)$ is sought, $a_\sigma$ is some given polynomial for each permutation $\sigma$ on $\{1,\dots,n\}$ and $f^\sigma(x_1,\dots,x_n) = f(x_{\sigma(1)}, \dots, x_{\sigma(n)})$.

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The situation $yf(x,y) - xf(y,x)$ is special: It is the case where $a_\sigma(x_1,\ldots, x_n) = a(x_{\sigma(1)},\ldots, x_{\sigma(n)})$. So $af$ would be a symmetric polynomial, and $f$ is just any symmetric polynomial divisible by $a$. – Willie Wong Nov 6 '12 at 11:21

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