The irreducibility of polynomials related to work of Heim, Luca and Neuhauser
Volume 210 / 2023
Abstract
Heim, Luca and Neuhauser (2019) introduced two families of polynomials in a variable $x$ generated by the arithmetic functions $g(n) = n$ and $g(n) = n^2$. They established the irreducibility over $\mathbb Q$ of the family generated by $g(n) = n$ and conjectured that the same held for the family generated by $g(n) = n^2$. Foster, Juillerat and Southwick (2018) confirmed this conjecture using Newton polygons. We generalise this work further, studying the corresponding family of polynomials generated by $g(n) = n^t$, where $t$ is a positive integer. We use properties of the Eulerian numbers to establish formulas for this family of polynomials and use Newton polygons to show that the family contains only irreducible polynomials over $\mathbb Q$.
