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

Bernhard Heim, Johann Stumpenhusen

TL;DR

The paper addresses the problem of detecting non-roots of D'Arcais polynomials $P_n^g(X)$, which encode Fourier coefficients of powers of the Dedekind eta function, by applying the Dedekind–Kummer theorem to localize roots at algebraic integers. The authors develop and deploy a framework that relates the splitting of $A_n^g(X)$ modulo primes to the arithmetic of algebraic integers $\alpha$, via minimal polynomials and indices of suborders, enabling new non-vanishing results. Their main contributions include extending non-root results to Gaussian integers and to algebro-number-theoretic settings in cyclotomic and quadratic fields for general $g$, with explicit modular conditions (e.g., $P_n^\sigma(ai+b)\neq0$ under $n\not\equiv5\pmod7$ and $21\nmid a$). Overall, the work broadens the toolkit for Lehmer-type questions by showing how local obstructions and index theory can yield concrete non-roots of $P_n^g(X)$ in richer arithmetic settings, potentially guiding further non-vanishing results and deeper structural understanding of these polynomials.

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.

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

TL;DR

The paper addresses the problem of detecting non-roots of D'Arcais polynomials , which encode Fourier coefficients of powers of the Dedekind eta function, by applying the Dedekind–Kummer theorem to localize roots at algebraic integers. The authors develop and deploy a framework that relates the splitting of modulo primes to the arithmetic of algebraic integers , via minimal polynomials and indices of suborders, enabling new non-vanishing results. Their main contributions include extending non-root results to Gaussian integers and to algebro-number-theoretic settings in cyclotomic and quadratic fields for general , with explicit modular conditions (e.g., under and ). Overall, the work broadens the toolkit for Lehmer-type questions by showing how local obstructions and index theory can yield concrete non-roots of in richer arithmetic settings, potentially guiding further non-vanishing results and deeper structural understanding of these polynomials.

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.

Paper Structure

This paper contains 12 sections, 12 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. D'Arcais polynomials.
  3. Previous work
  4. Results
  5. Preliminaries
  6. Algebraic number theory and the Dedekind--Kummer Theorem
  7. The index of a suborder
  8. The splitting of $A_n^g(X)$ modulo a prime $p$
  9. Proofs
  10. The form of $A_p^g(X)$ modulo $p$
  11. Integers on parallel lines of the real axis
  12. Open challenges

Key Result

Theorem 1.1

There is no pair of positive rational integers $(m,n)$ with $m \geq 3$ such that $\Phi_m(X) \mid P_n^\sigma(X)$.

Figures (1)

  • Figure 1: The Gaussian integers near the origin, each represented by an intersection of two lines. Previously known non-roots of each $P_n^\sigma(X)$ in black, newly found marked with empty points.

Theorems & Definitions (20)

  • Theorem 1.1: Heim--Luca--Neuhauser HeimLucaNeu*Theorem 1
  • Theorem 1.2: Żmija Zmija*Theorem 1
  • Theorem 1.3: Heim--Neuhauser HeimNeu25*Theorem 4
  • Theorem 1.4
  • Remark
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Dedekind--Kummer
  • Lemma 2.2: SteTall*Theorem 1.17
  • Lemma 2.3: Żmija Zmija*Lemma 5, Heim--Neuhauser HeimNeu25*Lemma 1
  • ...and 10 more