New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field
{"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}
引用次数: 0
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 $$d/2-\log _q d\le j\le d/2+\log _q d$$ .