On the Detection of Non-Roots of D'Arcais Polynomials

TL;DR

This paper extends algebraic number theory methods to identify more non-roots of D'Arcais polynomials, contributing to understanding their zero distribution and related conjectures in number theory.

Contribution

It generalizes the application of the Dedekind--Kummer Theorem to a broader class of D'Arcais polynomials, advancing the study of their non-zero Fourier coefficients.

Findings

Expanded the set of known non-roots of D'Arcais polynomials.

Applied Dedekind--Kummer Theorem to new polynomial classes.

Enhanced understanding of Fourier coefficient non-vanishing.

Abstract

The Lehmer conjecture states that the non-constant Fourier coefficients of the 24th power of the Dedekind eta function are non-zero. In a recent preprint, Neuhauser and the first author exploited an easily accessible tool from algebraic number theory, namely the Dedekind--Kummer Theorem, to prove the non-vanishing of the Fourier coefficients of certain powers of the Dedekind eta function at roots of unity. We extend the application of this method to enlarge the scope of non-roots of the related D'Arcais polynomials.