A Survey on Fixed Divisors

TL;DR

This survey comprehensively reviews the concept of fixed divisors of polynomials across various algebraic structures, highlighting key results, applications, and open questions to guide future research in algebra and number theory.

Contribution

It provides an exhaustive compilation of results on fixed divisors, explores their applications, and proposes directions for generalizing fixed divisors to matrix rings and other algebraic settings.

Findings

Compiled key results on fixed divisors across algebraic structures

Established connections between fixed divisors and integer-valued polynomials

Identified open problems and potential for generalization to matrix rings

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 $n \times n$ matrices over $\Z$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.