Stern polynomials and algebraic independence

1. Problem Statement

The paper addresses the problem of algebraic independence regarding specific values of power series derived from Stern polynomials.

• Context: The authors build upon the work of Dilcher and Eriksen (2018), who introduced "Type 1" Stern polynomials, denoted as $a_t(n; z)$. These are generalizations of the classical Stern diatomic sequence and the Stern polynomials introduced by Klavzar et al. (2007).
• Specific Object: The authors focus on a specific subsequence of these polynomials, $a_t(\alpha_n; z)$, where $\alpha_n = \frac{2^{kn} - (-1)^n}{2^k + 1}$. This subsequence converges to a power series $H_k(z)$ as $n \to \infty$.
• Core Question: For integers $t \ge 2$ and $k \ge 1$, are the values $H_k(\alpha)$ and $H_k(\alpha^{t^k})$ algebraically independent for any non-zero algebraic number $\alpha$ with $|\alpha| < 1$?

2. Methodology

The authors employ Mahler's method, a powerful technique in transcendental number theory used to prove the algebraic independence of values of functions satisfying specific functional equations.

The methodology proceeds through the following steps:

1. Functional Equations:
   The limit function $H_k(z)$ satisfies a linear functional equation derived from the recurrence of the Stern polynomials:
   $$H_k(z) = a_t(2k-1; z)H_k(z^{t^k}) + a_t(2k; z^{t^k})H_k(z^{t^{2k}})$$
   By defining $f_1(z) = H_k(z)$ and $f_2(z) = H_k(z^{t^k})$, the authors formulate a system of functional equations in matrix form:
   $$\begin{pmatrix} f_1(z^{t^k}) \\ f_2(z^{t^k}) \end{pmatrix} = A(z) \begin{pmatrix} f_1(z) \\ f_2(z) \end{pmatrix}$$
   where $A(z)$ is a matrix involving rational functions of $z$.

2. Application of Mahler's Theorem:
   The proof relies on Theorem 5.2 from Ku. Nishioka's book on Mahler's method. To apply this theorem, the authors must verify three conditions:

   • The functions $f_1(z)$ and $f_2(z)$ are transcendental over $\mathbb{C}(z)$.
   • The functions are algebraically independent over $\mathbb{C}(z)$.
   • The matrix $A(z)$ has no poles at the points $\alpha^{t^{nk}}$ for $n \ge 1$.

3. Auxiliary Lemmas:
   To satisfy the conditions for Mahler's theorem, the authors prove four key lemmas:

   • Lemma 2.1: Establishes that the only pole of the transformation matrix $A(z)$ is at $z=0$. Since the input $\alpha$ is non-zero and $|\alpha|<1$, the orbit $\alpha^{t^{nk}}$ never hits a pole.
   • Lemma 2.2: Proves that the ratio $H_k(1/2)/H_k(1/2^{t^k})$ is irrational. This is done by analyzing the continued fraction expansion of the ratio and showing it is an infinite regular continued fraction.
   • Lemma 2.3: Proves that $H_k(z)$ is transcendental over $\mathbb{C}(z)$. This uses Fatou's Theorem, which states that a power series with coefficients taking only finitely many values (here 0 and 1) is either rational or transcendental. The irrationality result from Lemma 2.2 rules out the rational case.
   • Lemma 2.4: Proves that $f_1(z)$ and $f_2(z)$ are algebraically independent over $\mathbb{C}(z)$. This is the most technical part, involving a contradiction argument. The authors assume dependence, construct a matrix product $G^{(n)}(z)$, and analyze the growth of coefficients and the behavior at $z=1$ to show that the assumed dependence leads to a contradiction (specifically, that coprime polynomials $a(z)$ and $b(z)$ must both vanish at $z=1$).

3. Key Contributions and Results

Main Theorem

Theorem 1.1: For any algebraic number $\alpha$ such that $0 < |\alpha| < 1$, the numbers $H_k(\alpha)$ and $H_k(\alpha^{t^k})$ are algebraically independent.

This means there is no non-zero polynomial $P(X, Y)$ with rational coefficients such that $P(H_k(\alpha), H_k(\alpha^{t^k})) = 0$.

Corollary on Continued Fractions

Corollary 1.1: The infinite continued fraction generated by the limit of the Stern polynomials:
$$\frac{H_k(\alpha)}{H_k(\alpha^{t^k})} = a_t(2k-1; \alpha) + \cfrac{a_t(2k; \alpha^{t^k})}{a_t(2k-1; \alpha^{t^k}) + \cfrac{a_t(2k; \alpha^{t^{2k}})}{a_t(2k-1; \alpha^{t^{2k}}) + \cdots}}$$
is transcendental for any algebraic $\alpha$ with $0 < |\alpha| < 1$.

Examples

The paper provides explicit examples for $k=1, 2, 3$ and general $t$. For instance, for $k=1$:
$$\frac{H_1(\alpha)}{H_1(\alpha^t)} = 1 + \cfrac{\alpha^t}{1 + \cfrac{\alpha^{t^2}}{1 + \cfrac{\alpha^{t^3}}{1 + \cdots}}}$$
The authors confirm these specific continued fractions yield transcendental numbers.

4. Significance

• Advancement in Transcendence Theory: The paper successfully extends Mahler's method to a new class of functions defined by Stern polynomials. It demonstrates the method's applicability to systems of functional equations arising from combinatorial sequences.
• Generalization of Known Results: While previous works (e.g., by Bundschuh, Väänänen, and Nishioka) have addressed the transcendence of continued fractions, this paper provides a unified framework for a broad family of "Type 1" Stern polynomials parameterized by $t$ and $k$.
• Connection to Combinatorics: The work bridges the gap between combinatorial number theory (Stern sequences) and analytic number theory (transcendence of values). It shows that the limits of these specific polynomial subsequences possess strong arithmetic properties (algebraic independence).
• Rigorous Proof of Independence: Proving algebraic independence is generally harder than proving transcendence. By establishing that the two values are independent, the authors provide a stronger result than merely stating the continued fraction is transcendental.

In summary, Duverney and Shiokawa have rigorously proven that the values of a specific class of power series derived from Stern polynomials, evaluated at algebraic points, generate algebraically independent numbers, thereby establishing the transcendence of the associated infinite continued fractions.