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arXiv · 2401.06964

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

Juan Francisco Gottig·Mariana Pérez·Melina Privitelli
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Abstract

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.

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Juan Francisco Gottig, Mariana Pérez, Melina Privitelli. 2024-01-13. An approach to the moments subset sum problem through systems of diagonal equations over finite fields. https://arxiv.org/abs/2401.06964

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