Reading through some math Olympiads it appears that there is a whole class of problems of this form:
Let $p_1,\ldots,p_n\in\mathbb {R}[x_1,\ldots,x_k] $ with $k,n\in \mathbb {N}^*$
Find all $f:\mathbb {R}\rightarrow\mathbb {R} $ such that $\sum_{i=1}^n f (p_i (x_1,\ldots,x_k))=0 $
Is there a general method, or approach for these problems? What happens when $f:\mathbb {N}\rightarrow \mathbb {N} $?
A trivial result is that $f (0)=0$ and heuristically is worth trying some values to evaluate the polynomials, looking at the roots of the polynomials or at the roots of $\sum_{i=1}^{n}{p_i(x_1,...,x_k)}$.
Can we expand on that?
Edit:since the above seems far too broad a case, we can ask the same questions in the case $\sum_{i=1}^n f (a_{i_1}x_1+...+a_{i_k}x_k)=0 $, with $a_{i_j}\in\mathbb {R} $