On polynomial representation functions for multivariate linear forms

Given an infinite sequence of positive integers A, we prove that, for every non-negative integer k, the number of solutions of the equation n=a1+⋯+ak, a1,…,ak∈A, is not constant for n sufficiently large. This result is a corollary of our main theorem, which partially answers a question of Sárközy and Sós on representation functions for multivariate linear forms. Additionally, we obtain an Erdős-Fuchs type result for a wide variety of representation functions.