Dynamical Canonical Heights and Finite Trees

Technical Summary: Dynamical Canonical Heights and Finite Trees

Problem Statement
This paper addresses the Dynamical Lehmer Conjecture, which predicts an asymptotically optimal lower bound for the canonical height $\hat{h}_f(\alpha)$ of a wandering algebraic point $\alpha$ in terms of its degree $[F(\alpha):F]$. Specifically, the authors seek to establish a lower bound of the form $\hat{h}_f(\alpha) \geq c / [F(\alpha):F]^2$ for specific classes of polynomials. This problem is the dynamical analogue of the classical Lehmer problem regarding the Mahler measure of integral polynomials. The authors focus on univariate polynomials $f$ that are monic, of prime power degree $d=p^e$, postcritically finite (all critical points are preperiodic), and hyperbolic (critical points lie in the Fatou set).

Methodology
The authors' approach is inspired by Dimitrov's proof of the Schinzel–Zassenhaus Conjecture, which utilized a finite topological tree (a "hedgehog") in the complex plane to bound the transfinite diameter of a set containing the conjugates of an algebraic integer. The authors adapt this strategy to the dynamical setting but introduce significant innovations to overcome limitations in their previous work [HS21].

1. Dynamical Trees and Hubbard Trees: Instead of using static hedgehogs, the authors construct a "dynamical hedgehog" based on the Hubbard tree $H_f$ of the postcritically finite polynomial $f$. They consider the sequence of preimages $H_f^{(s)} = (f^{(s)})^{-1}(H_f)$, which form nested finite topological trees.
2. Core Entropy and Convex Hulls: A central combinatorial tool is Thurston's core entropy $\rho(f)$. The authors prove that for hyperbolic polynomials not linearly conjugate to Chebyshev polynomials, the core entropy is strictly less than the degree ($\rho(f) < d$). This "non-maximal core entropy" property ensures that the convex hull of a sparse set of vertices in the preimage tree $H_f^{(s)}$ remains sparse (i.e., its size grows slower than the total size of the tree). This allows for the construction of a tree $T$ containing the Galois conjugates of a point with small height, with a controlled number of edges.
3. Transfinite Diameter Bounds: The authors develop new tools, including divided differences and estimates common in transcendence theory, to bound the transfinite diameter of these constructed trees from above. They avoid relying on Dubinin's Theorem (used in [HS21]) by deriving an upper bound directly using Vandermonde determinants and the combinatorial properties of the Hubbard tree.
4. Integrality and Congruences: To establish that the constructed tree has a transfinite diameter strictly less than 1 (implying the associated algebraic function is rational), the authors utilize Epstein's work on postcritically bounded maps. They prove that for polynomials of prime power degree with integral barycenters and postcritically bounded orbits, specific congruence relations hold modulo powers of $p$. These relations ensure that the power series expansion of a constructed algebraic function has integral coefficients.
5. Polya–Bertrandias Theorem: By combining the upper bound on the transfinite diameter with the integrality of the power series coefficients, the authors invoke the Pólya–Bertrandias Theorem. This forces the algebraic function to be a rational function, which imposes strong constraints on the original point $\alpha$, leading to the desired height bounds or preperiodicity.

Key Contributions and Results

• Lower Bounds for Canonical Heights: The paper establishes that for a monic, postcritically finite, hyperbolic polynomial $f \in \mathcal{O}_F[T]$ of prime power degree with an algebraic integer barycenter, there exists a constant $c > 0$ such that for any $f$-wandering algebraic integer $\alpha$:
  $$\hat{h}_f(\alpha) \geq \frac{c}{[F(\alpha):F]^2}$$
  This result removes the "Quill Hypothesis" required in the authors' previous work [HS21], applying to a broader class of polynomials.
• Local Height Bounds: The authors prove a Schinzel–Zassenhaus type lower bound for the local canonical height at archimedean places. If $\alpha$ is an $f$-wandering algebraic integer in a finite extension $L$, then:
  $$\max_{v \in M_\infty^L} \lambda_{f,v}(\alpha) \geq \frac{c(F,f)}{[L:F]}$$
• Bounds on Preperiod and Period Lengths: For $f$-preperiodic points, the authors derive explicit upper bounds on the preperiod length $\text{preper}(\alpha)$ and period length $\text{per}(\alpha)$ in terms of the field degree. Specifically, for polynomials not linearly conjugate to Chebyshev polynomials:
  $$\text{preper}(\alpha) \leq \left(1 + \left\lceil \frac{\log e}{\log d} \right\rceil \right) \left(1 + \frac{\log [F(\alpha):F]}{\log d}\right)$$
  $$\text{per}(\alpha) \leq p^f [F(\alpha):F]$$
  where $e$ and $f$ relate to the ramification and residue degrees of primes above $p$.
• Combinatorial Analysis of Hubbard Trees: The paper provides a detailed analysis of the growth of the Hubbard tree under iteration, linking the spectral radius of the induced map on edges (core entropy) to the sparsity of convex hulls in preimage trees. This combinatorial insight is crucial for bounding the transfinite diameter without relying on external deep theorems like Dubinin's.
• Examples and Limits: The authors provide examples of polynomials to which their results apply, including $(T^2-1)^2$. They also discuss the limits of the analogy between postcritically finite polynomials and elliptic curves with complex multiplication (CM), noting that while preperiodic points in both settings share Galois-theoretic properties, postcritically finite polynomials do not necessarily possess good reduction everywhere, unlike CM elliptic curves.

Significance and Claims
The authors claim that this work provides new evidence towards the Dynamical Lehmer Conjecture for univariate polynomials. By replacing the static hedgehog construction with a dynamical tree derived from the Hubbard tree, and by developing elementary yet powerful tools to bound transfinite diameters, they extend the scope of the Schinzel–Zassenhaus method to a wider class of dynamical systems. The removal of the Quill Hypothesis is a significant technical advancement. The paper positions itself as a bridge between arithmetic dynamics, complex dynamics (via Hubbard trees and core entropy), and transcendence theory (via integrality and transfinite diameter bounds). The authors modestly note that their results apply to specific classes of polynomials (prime power degree, hyperbolic, postcritically finite) and that future work will aim to address subhyperbolic cases and more general polynomials.