Cyclotomic factors of rational necklace functions

Technical Summary: Cyclotomic Factors of Rational Necklace Functions

Problem Statement
The paper addresses the arithmetic properties of necklace polynomials, specifically their factorization into cyclotomic polynomials. While necklace polynomials ($M_n(x)$) and Fekete polynomials ($F_n(x)$) have been studied extensively in combinatorics and number theory, their cyclotomic factors are not fully understood. Previous work by Hyde [9] and the authors themselves [3] utilized Mahler algebras and combinatorial arguments to identify certain cyclotomic factors. However, numerical data revealed that these existing frameworks fail to explain the appearance of specific cyclotomic factors in both classical necklace polynomials and Fekete polynomials. The central problem is to unify these two families of polynomials and provide a comprehensive theoretical framework that explains all observed cyclotomic factors, including those previously unexplained.

Methodology
The authors introduce a generalized framework of rational necklace functions. For a rational function $f \in \mathbb{Q}(x)$, the $n$-th rational necklace function is defined as:
$$M_n(f) = \sum_{d|n} \mu(d) f(x^d) = \psi_n f$$
where $\psi_n = \sum_{d|n} \mu(d)[d]$ is an element of the Mahler algebra $\mathbb{Z}[\mathbb{Z}^\circ]$. This algebra acts on rational functions via $[m] \cdot f(x) = f(x^m)$.

The methodology proceeds through three main theoretical layers:

1. Mahler Algebra and Symmetries: The authors analyze the action of elements in the Mahler algebra on rational functions. They establish that cyclotomic factors $\Phi_d$ arise when the algebraic element $\alpha$ (such as $\psi_n$) satisfies specific congruence conditions modulo $[d]$ (i.e., $\phi_d(\alpha) = 0$ in the quotient ring) and when the function $f$ possesses specific symmetries (e.g., $f(x) + f(1/x) = C$).
2. Signed Mahler Algebra: To address symmetries involving negative signs (e.g., $f(x) + f(-x) = 0$ or $f(x) + f(-1/x) = 0$), the authors introduce a signed Mahler algebra $R_\pm = \mathbb{Z}[\{\pm 1\} \times \mathbb{Z}_{odd}]$. This extension allows for the description of functional equations that the standard Mahler algebra cannot capture, particularly those relevant to Fekete polynomials.
3. Galois Symmetries and Field of Definition: The paper identifies a "lowering field of definition" phenomenon. By analyzing the interaction between the Galois group of cyclotomic fields $\text{Gal}(\mathbb{Q}(\zeta_d)/\mathbb{Q})$ and the Mahler algebra elements, the authors show that if a rational necklace function takes values in a subfield fixed by a specific Galois subgroup, additional cyclotomic factors can appear. This relies on the compatibility between the Galois action ($\zeta \mapsto \zeta^a$) and the Mahler action ($f(x) \mapsto f(x^a)$).

Key Contributions and Results

• Unification of Families: The paper unifies necklace polynomials and Fekete polynomials under the single definition of rational necklace functions $M_n(f)$. It demonstrates that $M_n(f)$ recovers classical necklace polynomials (up to a scalar) when $f(x) = x$ and modified Fekete polynomials when $f(x) = \frac{x}{1-x}$.
• Generalized Factorization Theorems:
  • Theorem 2.7: Establishes that if $\phi_d(\alpha) = 0$ and $f$ is defined at relevant roots of unity, then $\Phi_d$ is a factor of $\alpha f$. This generalizes Hyde's results.
  • Theorem 2.14 & 2.18: Connects the existence of cyclotomic factors to the annihilator ideal of $f$. If $f$ has a symmetry (e.g., $f(x) + f(1/x) = 0$) and $\alpha$ is congruent to an element in the ideal generated by that symmetry modulo $[d]$, then $\Phi_d$ divides $\alpha f$. This explains why Fekete polynomials possess more factors than classical necklace polynomials.
  • Theorem 3.6 & 3.10: Utilizes the signed Mahler algebra to explain factors of the form $\Phi_{2m}$ arising from symmetries like $f(x) + f(-x) = 0$.
• Discovery of Galois-Driven Factors:
  • Theorem 4.5: The authors identify a new source of cyclotomic factors arising from the interaction between the Galois group and the Mahler algebra. Specifically, if $\alpha_1 f(\zeta_d)$ lies in a subfield $\mathbb{Q}(\zeta_{d_1})$ due to Galois symmetries, and $\alpha_2$ satisfies specific congruence conditions modulo $[d_1]$, then $\Phi_d$ divides $\alpha_1 \alpha_2 f$.
  • This phenomenon explains factors such as $\Phi_{24}$ in $M_{3 \times 7 \times p}(f)$ for primes $p \equiv 1 \pmod 8$, which were previously unexplained by combinatorial pairing or standard symmetry arguments.

Significance
The paper claims to provide a "general framework" that unifies and generalizes earlier work on necklace and Fekete polynomials. Its primary significance lies in:

1. Conceptual Explanation: It offers a conceptual reason for the "surprising phenomenon" where Fekete polynomials have more cyclotomic factors than necklace polynomials, attributing this to the specific symmetries of the underlying rational function $f$.
2. New Mechanism: It uncovers a "lowering field of definition" phenomenon driven by Galois groups, revealing a hidden source of cyclotomic factors that were missed by previous combinatorial approaches.
3. Algebraic Tools: The development of the signed Mahler algebra and the detailed study of its ring-theoretic properties (including zero-divisors and annihilator ideals) provide new tools for studying necklace-like functions, such as dynatomic polynomials.

The authors maintain a modest tone, noting that while their approach explains many previously unexplained factors, the framework is designed to open new avenues for studying similar functions rather than providing a complete classification of all possible factors. The results are presented as theoretical generalizations of existing theorems in [3], [5], and [9], rather than as experimental discoveries of new applications.