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 = a 1 + ⋯ + a k , a 1 , … , a k ∈ 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.