  Let $\mathbb{F}_q$ be the finite field of $q$ elements, for a given subset
$D\subset \mathbb{F}_q$, $m\in \mathbb{N}$, an integer $k\leq |D|$ and
$\boldsymbol{b}\in \mathbb{F}_q^m$ we are interested in determining the
existence of a subset $S\subset D$ of cardinality $k$ such that $\sum_{a\in
S}a^i=b_i$ for $i=1,\ldots, m$. This problem is known as the moment subset sum
problem and it is $NP$-complete for a general $D$. We make a novel approach of
this problem trough algebraic geometry tools analyzing the underlying variety
and employing combinatorial techniques to estimate the number of
$\mathbb{F}_q$-rational points on certain varieties. We managed to give
estimates on the number of $\mathbb{F}_q$-rational points on certain diagonal
equations and use this results to give estimations and existence results for
the subset sum problem.
