A Survey on Fixed Divisors

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 $\mathbb{Z},$ 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 $\mathbb{Z}$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.

Accepted for publication in Confluentes Mathematici

References in corpus (10)

• Integer-valued polynomials on algebras
• A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
• Integral-valued polynomials over the set of algebraic integers of bounded degree
• Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring
• Integer-valued polynomials over matrices and divided differences
• Integral closure of rings of integer-valued polynomials on algebras
• Factorization of integer-valued polynomials with square-free denominator
• Galois structure on integral valued polynomials
• Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power
• Simultaneous $p$-orderings and minimising volumes in number fields

Cited by in corpus (1)

• Irreducibility of integer-valued polynomials I