A Survey on Fixed Divisors

Q4 Mathematics

Confluentes Mathematici Pub Date : 2017-09-23 DOI:10.5802/cml.54

Confluentes Mathematici, Devendra Prasad, K. Rajkumar, Satyanarayana Reddy, Devendra Prasad, K. Rajkumar

{"title":"A Survey on Fixed Divisors","authors":"Confluentes Mathematici, Devendra Prasad, K. Rajkumar, Satyanarayana Reddy, Devendra Prasad, K. Rajkumar","doi":"10.5802/cml.54","DOIUrl":null,"url":null,"abstract":"In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n×n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting. keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2017-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Confluentes Mathematici","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/cml.54","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}

引用次数: 5

Abstract

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n×n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting. keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials

查看原文本刊更多论文

关于固定除数的综述

在这篇文章中，我们汇编了许多数学家关于多项式的固定因子的研究成果。本文简明扼要地解释了大多数结果，并打算进行详尽的调查。我们给出了固定因子在各种代数环境中的结果，以及固定因子在各种代数和数论问题中的应用。这项工作以有序的方式呈现，以便从Z的最简单情况开始，逐步导致Dedekind域的情况。我们也会根据上下文提出一些开放的问题，这可能会促使读者在这个方向上进一步努力。我们描述了整数多项式环中固定因子的各种界以及固定因子与不同概念的联系。最后，我们建议在Z(或Dedekind定义域)上n×n矩阵环的情况下，整数多项式环的泛化如何导致固定因子在该设置中的泛化。关键词固定因子，广义阶乘，多变量广义阶乘，指标公因子，素数理想因子分解，整值多项式

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Confluentes Mathematici Mathematics-Mathematics (miscellaneous)

CiteScore

0.60

自引率

0.00%

发文量

期刊介绍： Confluentes Mathematici is a mathematical research journal. Since its creation in 2009 by the Institut Camille Jordan UMR 5208 and the Unité de Mathématiques Pures et Appliquées UMR 5669 of the Université de Lyon, it reflects the wish of the mathematical community of Lyon—Saint-Étienne to participate in the new forms of scientific edittion. The journal is electronic only, fully open acces and without author charges. The journal aims to publish high quality mathematical research articles in English, French or German. All domains of Mathematics (pure and applied) and Mathematical Physics will be considered, as well as the History of Mathematics. Confluentes Mathematici also publishes survey articles. Authors are asked to pay particular attention to the expository style of their article, in order to be understood by all the communities concerned.