Bhargava factorials and irreducibility of integer-valued polynomials

TL;DR

This paper uses Bhargava factorials to analyze the irreducibility of integer-valued polynomials over subsets of integers and extends the results to subsets of Dedekind domains.

Contribution

It introduces a novel application of Bhargava factorials for irreducibility testing of integer-valued polynomials and generalizes to Dedekind domains.

Findings

Bhargava factorials effectively determine polynomial irreducibility.

Generalization of irreducibility criteria to Dedekind domain subsets.

New methods for analyzing factorization in integer-valued polynomial rings.

Abstract

The ring of integer-valued polynomials over a given subset $S$ of $\Z$ (or $ \mathrm{Int}(S,\Z ))$ is defined as the set of polynomials in $\Q[x]$ which maps $S$ to $\Z$. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial $f \in \mathrm{Int}(S,\Z ))$. We also generalize our results to arbitrary subsets of a Dedekind domain.