Recently, I have come across following problem. The origin is the following: we study a set of $q^2$ points in $AG(3,q)$ with some special properties. Using the polynomial method (the full derivation is quite long, so I won't copy it here), we find two sets of multisets of elements of $GF(q)$ (coming from their coordinates) satisfying certain equations. In order to extract more information from the main polynomial under consideration, we need more relations on the coefficients, which are combinations of elements of these two sets. For the details, see below.
The two (multi)sets of elements of $GF(q)$ consist both of $q^2$ elements, say $\{a_i\}_{i=1}^{q^2}$ and $\{b_i\}_{i=1}^{q^2}$. For these two sets it is given that (which comes from the fact that some coefficients of the main polynomial are zero):
\begin{align*} \sum_{i=1}^{q^2} a_i^rb_i^s = 0 \end{align*}
for all $r,s$ such that $0 \leq r,s \leq q-1$ and $r + s \leq q$. Is there anything we can infer for higher degree, i.e., when the sum $r + s > q$ ? This would imply that other terms also vanish in the main polynomial, from which we can hopefully derive information on the set of $q^2$ points.
For example, I'd like to know if also $\sum_{i=1}^{q^2} a_i^{q-2}b_i^3$ equals zero.