On polynomial representation functions for multilinear forms

Given an infinite sequence of positive integers $\cA$, we prove that for every nonnegative integer $k$ the number of solutions of the equation $n=a_1+...+a_k$, $a_1,\,..., a_k\in \cA$, is not constant for $n$ large enough. This result is a corollary of our main theorem, which partially answers a question of Sárközy and Sós on representation functions for multilinear forms. Additionally, we obtain an Erdős-Fuchs type result for a wide variety of representation functions.

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