On polynomial representation functions for multilinear forms

Given an infinite sequence of positive integers $\text{cA}$, we prove that for every nonnegative integer $k$ the number of solutions of the equation $n=a_1+\mathrm{.}\mathrm{.}\mathrm{.}+a_k$, $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k\in \text{cA}$, is not constant for $n$ large enough. This result is a corollary of our main theorem, which partially answers a question of S'ark"ozy and S'os on representation functions for multilinear forms. Additionally, we obtain an Erd\H{o}s-Fuchs type result for a wide variety of representation functions.