{"title":"On the stable polynomials of degrees 2,3,4","authors":"Tong Lin, Qiang Wang","doi":"10.1016/j.ffa.2024.102474","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>q</em> be a prime power. For <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>, we construct stable polynomials of the form <span><math><msup><mrow><mi>b</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>c</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>d</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> by Capelli's lemma. Moreover, when <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we improve a lower bound for the number of stable quadratic polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> due to Goméz-Pérez and Nicolás <span><span>[4]</span></span>. When <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span>, we prove Ahmadi and Monsef-Shokri's conjecture <span><span>[2]</span></span> that <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> is stable over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102474"},"PeriodicalIF":1.2000,"publicationDate":"2024-08-08","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/S1071579724001138","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let q be a prime power. For , we construct stable polynomials of the form over by Capelli's lemma. Moreover, when and , we improve a lower bound for the number of stable quadratic polynomials over due to Goméz-Pérez and Nicolás [4]. When , we prove Ahmadi and Monsef-Shokri's conjecture [2] that is stable over .
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.
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