{"title":"有限域上不可约多项式和规定系数的自倒不可约多项式的新估计和存在性结果","authors":"Zhicheng Gao","doi":"10.1007/s44007-023-00062-1","DOIUrl":null,"url":null,"abstract":"A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${\\mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-\\log _q d\\le j\\le d/2+\\log _q d$$ .","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field\",\"authors\":\"Zhicheng Gao\",\"doi\":\"10.1007/s44007-023-00062-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${\\\\mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-\\\\log _q d\\\\le j\\\\le d/2+\\\\log _q d$$ .\",\"PeriodicalId\":74051,\"journal\":{\"name\":\"La matematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"La matematica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s44007-023-00062-1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"La matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s44007-023-00062-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${\mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-\log _q d\le j\le d/2+\log _q d$$ .