Juan Francisco Gottig , Mariana Pérez , Melina Privitelli
{"title":"An approach to the moments subset sum problem through systems of diagonal equations over finite fields","authors":"Juan Francisco Gottig , Mariana Pérez , Melina Privitelli","doi":"10.1016/j.ffa.2024.102511","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> be the finite field of <em>q</em> elements. Let <em>m</em> and <em>k</em> be positive integers, <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> and <span><math><mi>D</mi><mo>⊂</mo><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> with <span><math><mi>k</mi><mo>≤</mo><mo>|</mo><mi>D</mi><mo>|</mo></math></span>. Our aim is to determine the existence of a subset <span><math><mi>S</mi><mo>⊂</mo><mi>D</mi></math></span> of cardinality <em>k</em> such that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>S</mi></mrow></msub><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi></math></span>. This problem is known as the <em>moments subset sum problem</em>. We take a novel approach to this problem by establishing a connection between the existence of such a subset <em>S</em> 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 <em>moments subset sum problem</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102511"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001503","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be the finite field of q elements. Let m and k be positive integers, and with . Our aim is to determine the existence of a subset of cardinality k such that for . 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.
设 Fq 是有 q 个元素的有限域。设 m 和 k 为正整数,b∈Fqm,D⊂Fq,k≤|D|。我们的目标是确定是否存在一个心数为 k 的子集 S⊂D,使得 i=1,...m 时,∑a∈Sai=bi。这个问题被称为矩子集和问题。我们对这一问题采取了一种新的方法,即在这样一个子集 S 的存在与确定某个对角方程组是否至少有一个具有不同坐标的有理解这一问题之间建立联系。为此,我们研究了上述系统解集的一些相关几何性质。通过分析,我们可以对对角方程组的有理解的数量进行估计,随后我们将应用这些结果来解决矩子集和问题。
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.