{"title":"Some classes of permutation pentanomials","authors":"Zhiguo Ding , Michael E. Zieve","doi":"10.1016/j.ffa.2025.102619","DOIUrl":null,"url":null,"abstract":"<div><div>For each prime <span><math><mi>p</mi><mo>≠</mo><mn>3</mn></math></span> and each power <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, we present two large classes of permutation polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of the form <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>B</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> which have at most five terms, where <span><math><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a polynomial with coefficients in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. The special case <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102619"},"PeriodicalIF":1.2000,"publicationDate":"2025-03-27","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/S1071579725000498","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
For each prime and each power , we present two large classes of permutation polynomials over of the form which have at most five terms, where is a polynomial with coefficients in . The special case of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case , no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.
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.