Polynomials defining many units

TL;DR

This paper classifies polynomials with integer coefficients that produce units in integral group rings for infinitely many group element orders, linking them to generic units and roots of fixed integers.

Contribution

It provides a complete classification of such polynomials, connecting their properties to the concept of generic units and roots of fixed integers.

Findings

Polynomials defining units for infinitely many orders are exactly the generic units.

Classification of polynomials that yield units on roots of a fixed integer for infinitely many n.

Establishes a clear criterion linking polynomial properties to units in group rings.

Abstract

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if the polynomial defines generic units in the sense of Marciniak and Sehgal. We also classify the polynomials with integral coefficients which provides units when evaluated on $n$-roots of a fixed integer $a$ for infinitely many positive integers $n$.