An approach to the moments subset sum problem through systems of diagonal equations over finite fields

Let $F_q$ be the finite field of q elements. Let m and k be positive integers, $b\in F_q^m$ and $D\subset F_q$ with $k\le \left|D\right|$. Our aim is to determine the existence of a subset $S\subset D$ of cardinality k such that ${\sum }_{a\in S}{a}^i=b_i$ for $i=1,\dots ,m$. This problem is known as the moments subset sum problem. We take a novel approach to this problem by establishing a connection between the existence of such a subset S with the problem of determining if a certain system of diagonal equations has at least one rational solution with distinct coordinates. To achieve this, we study some relevant geometric properties of the set of solutions of the mentioned system. This analysis allows us to provide estimates on the number of rational solutions of systems of diagonal equations and we subsequently apply these results to address the moments subset sum problem.