An approach to the moments subset sum problem through systems of diagonal equations over finite fields
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| Format: | Preprint |
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2024
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| author | Gottig, Juan Francisco Pérez, Mariana Privitelli, Melina |
| author_facet | Gottig, Juan Francisco Pérez, Mariana Privitelli, Melina |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An approach to the moments subset sum problem through systems of diagonal equations over finite fields Gottig, Juan Francisco Pérez, Mariana Privitelli, Melina Number Theory Combinatorics 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. |
| title | An approach to the moments subset sum problem through systems of diagonal equations over finite fields |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2401.06964 |