A Survey on Fixed Divisors

Prasad, Devendra; Rajkumar, Krishnan; Reddy, A. Satyanarayana

Confluentes Mathematici, Tome 11 (2019) no. 1, pp. 29-52.

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 a Dedekind domain) could lead to the generalization of fixed divisors in that setting.

Classification : 11Sxx, 11S05, 13F20
Mots clés : Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials

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