What question did this study set out to answer?

The central aim is to explore the non-vanishing Fourier coefficients of D’Arcais polynomials using established number theory techniques.

April 12, 2026Research in Number Theory0 citationsOpen Access

On the detection of non-roots of D’Arcais polynomials

BHBernhard Heim JSJohann Christian Stumpenhusen

Key Points

The central aim is to explore the non-vanishing Fourier coefficients of D’Arcais polynomials using established number theory techniques.
Utilized the Dedekind-Kummer Theorem from algebraic number theory.
Applied this theorem to analyze the Fourier coefficients of certain powers of the Dedekind eta function.
Expanded the investigation to include non-roots of D’Arcais polynomials.
Confirmed the non-vanishing of Fourier coefficients at roots of unity.
Broadened the understanding of D’Arcais polynomials beyond previously established results.

Abstract

Abstract The Lehmer conjecture states that the non-constant Fourier coefficients of the 24th power of the Dedekind eta function are non-zero. Recently, 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.

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