On polynomial representation functions for multilinear forms

Abstract

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 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.