Arithmetic exceptionality of generalized Chebyshev polynomials of the second kind

Technical Summary: Arithmetic Exceptionality of Generalized Chebyshev Polynomials of the Second Kind

Problem Statement
The paper investigates the arithmetic exceptionality of generalized Chebyshev polynomials of the second kind, denoted $Q^k_{\mathfrak{g}}$, associated with semi-simple Lie algebras $\mathfrak{g}$. A polynomial $f \in \mathbb{Z}[x]$ in $n$ variables is defined as arithmetically exceptional if the induced map $f: \mathbb{F}_p^n \to \mathbb{F}_p^n$ is a permutation for infinitely many primes $p$.

While the arithmetic exceptionality of the first kind generalized Chebyshev polynomials ($P^k_{\mathfrak{g}}$) is well-understood (e.g., $P^k_{A_1}$ corresponds to Dickson polynomials of the first kind, which are exceptional for certain $k$), the behavior of the second kind ($Q^k_{\mathfrak{g}}$) is less clear. Specifically, for the classical case $\mathfrak{g} = A_1$, it is known that $Q^k_{A_1}$ (equivalent to Dickson polynomials of the second kind $E_k(x,1)$) is not arithmetically exceptional for $k \ge 2$. The authors address the open problem of whether this non-exceptionality extends to higher-rank Lie algebras, focusing specifically on the rank-two case $\mathfrak{g} = A_2$. The polynomials $Q^k_{A_2}$ are bivariate mappings defined via the characters of irreducible representations of $A_2$.

Methodology
The authors employ a combination of algebraic number theory, the theory of finite fields, and the representation theory of Lie algebras. Their approach relies on the following key mechanisms:

1. Diagonal Restriction: To analyze the bivariate map $Q^k_{A_2}: \mathbb{F}_q^2 \to \mathbb{F}_q^2$, the authors restrict the domain to the diagonal $D = \{(a, a) \mid a \in \mathbb{F}_q\}$. They define a univariate polynomial $f_k(x) = h_k(x, x)$, where $Q^k_{A_2} = (h_k(y_1, y_2), h_k(y_2, y_1))$. They establish that if $Q^k_{A_2}$ is a permutation, then its diagonal restriction $f_k$ must also be a permutation. This reduces the problem to the univariate case.
2. Cyclotomic Parametrization: Utilizing a correspondence between the finite field $\mathbb{F}_q$ and the fixed points of the first kind generalized Chebyshev map $P^q_{A_1}$, the authors parametrize elements of $\mathbb{F}_q$ using values of the form $2\cos(2\pi u)$ derived from roots of unity. This allows them to translate the permutation problem into questions about the behavior of trigonometric sums and cyclotomic elements.
3. Norm Analysis: The core of the proof involves calculating the field norms of specific cyclotomic elements. By analyzing the product of the images of non-zero elements under the map $f_k$, the authors derive conditions under which the resulting polynomial cannot be a permutation. They utilize properties of the norm map in cyclotomic extensions $\mathbb{Q}(\zeta_n)$ and their maximal real subfields.
4. Congruence Conditions: The authors derive necessary conditions involving greatest common divisors (gcd) and modular arithmetic constraints on $k$ and the prime $p$ for $f_k$ to be a permutation. They show that for sufficiently large primes $p$, these conditions fail.

Key Contributions and Results
The paper provides a rigorous proof that the generalized Chebyshev polynomials of the second kind associated with the root system $A_2$ are not arithmetically exceptional.

• Main Theorem (Theorem 24): For any integer $k > 1$, the bivariate polynomial mapping $Q^k_{A_2}: \mathbb{F}_q^2 \to \mathbb{F}_q^2$ is not a permutation for all but finitely many primes $p$. Specifically, if $Q^k_{A_2}$ is a permutation, then $p < |\tau(k)| + 1$, where $\tau(k)$ is a function defined by the authors involving $k$. Consequently, $Q^k_{A_2}$ is not arithmetically exceptional.
• Univariate Case ($A_1$): The paper revisits Cohen's theorem regarding $Q^k_{A_1}$ (Dickson polynomials of the second kind), providing a self-contained proof that $Q^k_{A_1}$ is not arithmetically exceptional for $k \ge 2$ by establishing a bound on $p$ based on $k$.
• Recurrence Relations: The authors explicitly define the recurrence relations for the components of $Q^k_{A_2}$, noting that while they share the same recurrence as the first kind polynomials $P^k_{A_2}$, the distinct initial conditions lead to fundamentally different arithmetic behaviors.
• Explicit Formulas: The paper derives explicit formulas for the values of the diagonal restriction $f_k$ at specific points (e.g., $0, 1, 2, 3, -1$) and establishes the functional equation for $f_k$ in terms of roots of unity (Theorem 29).

Significance and Claims
The authors claim that this work extends the understanding of arithmetic exceptionality from the classical univariate case ($A_1$) to the first non-trivial multivariate case ($A_2$). They highlight that while the first kind polynomials $P^k_{A_2}$ are exceptional for infinitely many primes (as shown in previous literature), the second kind polynomials $Q^k_{A_2}$ exhibit the same non-exceptional behavior as their $A_1$ counterparts.

The paper concludes by noting that the methods developed—specifically the reduction to the diagonal and the use of cyclotomic norms—could potentially be generalized to other semi-simple Lie algebras, such as $B_2$ and $G_2$. However, the authors modestly acknowledge that immediate difficulties arise in these cases because the symmetry property allowing the diagonal restriction (which holds for $A_2$) does not necessarily hold for other root systems. The work is presented as a foundational step in classifying the arithmetic properties of these generalized polynomial families.