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Local height arguments toward the dynamical Mordell-Lang conjecture

This paper proves that regular endomorphisms of complex affine space with a sufficiently large degree gap satisfy the dynamical Mordell-Lang conjecture for curves, a result achieved by demonstrating that such a gap forces all periodic curves to be vertical lines passing through the origin.

Original authors: She Yang, Aoyang Zheng

Published 2026-05-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: She Yang, Aoyang Zheng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a complex dance performed by points on a stage. This stage is a mathematical space called "affine space" (think of it as an infinite grid). The dance is choreographed by a rule called an endomorphism (a function that moves every point to a new spot).

The Dynamical Mordell–Lang Conjecture is a big question about this dance: If you pick a starting point and a specific path (a curve) on the stage, will the dancer visit that path again and again in a predictable pattern? The conjecture says yes: the times they visit will form a neat list of repeating intervals (like visiting every 3rd day, or every 5th day), not a chaotic mess.

This paper, by She Yang and Aoyang Zheng, tackles a specific version of this problem where the dance rules have a special "gap" in their complexity. Here is the breakdown of their findings using everyday analogies:

1. The "Gap" in the Dance Rules

The authors look at a specific type of dance rule where the movement is mostly determined by a "main beat" (high-degree polynomials) and a "background rhythm" (lower-degree polynomials).

The Main Beat: Imagine the dancers are mostly moving based on a powerful, dominant force (like a strong wind).
The Gap: The authors require that the "background rhythm" is significantly weaker than the main beat. They call this a degree gap.
The Condition: They prove that if this gap is large enough (specifically, if the gap is more than twice the "complexity" of how the dance behaves at the edge of the stage), then the conjecture holds true.

2. The Edge of the Stage (Infinity)

In math, we often imagine the stage has an "edge" or a horizon called infinity ( $H_\infty$ ).

The dance rules extend to this edge. The authors look at how the dancers behave right at the horizon.
They check if the dancers get "stuck" or "multiply" in a complicated way at the edge (this is called ramification or multiplicity).
The Rule: If the "gap" in the dance rules is bigger than twice the maximum number of times a dancer gets stuck at the edge, everything works out nicely.

3. The Big Discovery: "Vertical Lines"

The most surprising part of their proof is what happens to the paths (curves) that the dancers visit infinitely often.

The Finding: Under their strict "gap" conditions, any path that gets visited infinitely often turns out to be a straight line that goes directly through the center of the stage (the origin).
The Metaphor: Imagine the dancers are running on a giant field. You might expect them to run in circles, figure-eights, or wild spirals. But the authors prove that if the "gap" condition is met, the only paths they can run on forever are straight lines shooting out from the center.
They call these "vertical lines" because, in their mathematical coordinate system, they stand straight up relative to the horizon.

4. Why This Matters (The "Local Height" Argument)

To prove this, the authors used a tool called Local Height.

The Analogy: Think of "height" as a measure of how "far out" or "complicated" a point is.
The Problem: Usually, in these types of problems, points grow in complexity at the same speed everywhere (like a balloon inflating evenly). This makes it hard to distinguish between different paths.
The Solution: The authors' "gap" condition breaks this symmetry. It creates two different speeds of growth:
1. The main movement grows very fast.
2. The movement near the edge (infinity) grows at a different, controlled speed.
By using a famous mathematical theorem (Roth's Theorem) to compare these speeds, they showed that any curve trying to wiggle or bend would eventually break the rules. The only shape that survives is the straight line.

5. Is the Rule Perfect?

The authors also checked if their "gap" rule was the best possible.

The Result: Yes, it is optimal. They provided examples showing that if you make the gap just a tiny bit smaller (changing "greater than" to "greater than or equal to"), the magic breaks. The dancers can then run on wild, non-straight paths. This proves their condition is the exact tipping point needed to force the paths to be straight lines.

Summary

In simple terms, the authors proved that for a specific class of mathematical dances where the "main beat" is much stronger than the "background noise," any path that gets visited infinitely often must be a straight line shooting from the center. This confirms the Dynamical Mordell–Lang conjecture for these specific cases and reveals a hidden simplicity (straight lines) in what could have been a chaotic system.

Technical Summary: Local Height Arguments toward the Dynamical Mordell–Lang Conjecture

Problem Statement
This paper investigates the Dynamical Mordell–Lang (DML) conjecture for regular endomorphisms of the complex affine space $\mathbb{A}^N_{\mathbb{C}}$ . The DML conjecture posits that for an endomorphism $f$ of a quasi-projective variety $X$ , the return set $\{n \in \mathbb{N} \mid f^n(x) \in V\}$ of a point $x$ into a closed subvariety $V$ is a finite union of arithmetic progressions. While the conjecture is known for specific cases (e.g., étale endomorphisms or $\mathbb{A}^2$ ), this work focuses on the case where $V$ is a curve (the DML(1) property) for a specific class of regular endomorphisms with a "degree gap."

The authors consider regular endomorphisms $f: \mathbb{A}^N \to \mathbb{A}^N$ that extend to $\mathbb{P}^N$ . Such maps are of the form $f(x) = (f_1+g_1, \dots, f_N+g_N)$ , where $f_i$ are homogeneous polynomials of degree $d$ with no common zeros, and $g_i$ are polynomials of degree $\le d-k$ . The integer $k$ is the "degree gap." The behavior of the induced map $f_\infty$ on the hyperplane at infinity $H_\infty$ is crucial. The central problem is to determine under what conditions on $k$ and the ramification of $f_\infty$ the DML(1) property holds and to characterize the periodic curves.

Methodology
The authors employ a "local height argument," a technique distinct from the global height arguments used in previous works (e.g., by Ghioca, Tucker, and Xie) which rely on the system being "polarized" in a way that allows for two distinct growth speeds. Since the systems studied here are polarized, the global height argument fails. Instead, the authors utilize:

Arakelov Theory for Arithmetic Function Fields: To work over $\mathbb{C}$ , the authors descend to an arithmetic function field $K$ (a finitely generated extension of $\mathbb{Q}$ ) where all data are defined. They utilize the Arakelov theory developed by Vojta and Moriwaki, which provides a product formula and a measure space of absolute values $M_K$ .
Local Heights (Weil Functions): They define Weil functions $\lambda_P$ for points and $\lambda_{H_\infty}$ for the hyperplane at infinity over the set of places $S$ . These functions measure the proximity of points to specific subvarieties.
Roth's Theorem: A central tool is Roth's theorem over arithmetic function fields (established by Vojta). The authors use this to control the distribution of points on curves. Specifically, they prove that if a curve is "sufficiently vertical" (meaning points on the curve approach the hyperplane at infinity in a specific manner relative to their projection), the curve must be a straight line.
Northcott's Finiteness Property: This is used to establish the finiteness of periodic points with bounded height, ensuring that the backward orbits of points at infinity are finite sets of periodic points.
Triangle Inequality and Iteration: The proof involves constructing a sequence of pre-images of the curve $C$ (denoted $C_{-n}$ ) and showing that under the degree gap condition, these curves become increasingly "vertical" at their intersection points with $H_\infty$ .

Key Contributions and Results

Main Theorem (Theorem 1.3): Let $f$ be a regular endomorphism of $\mathbb{A}^N$ with degree gap $k$ . If $C$ is an irreducible closed subcurve with an infinite intersection with an orbit of $f$ , and if $k > 2 \max_{P \in I_0} e_{f_\infty}(P)$ (where $I_0$ is the forward orbit of the intersection points $C \cap H_\infty$ and $e_{f_\infty}(P)$ is the multiplicity of $f_\infty$ at $P$ ), then:
1. The intersection $C \cap H_\infty$ consists of 1 or 2 points, which are periodic under $f_\infty$ .
2. $C$ is a "vertical line" (a straight line passing through the origin).
DML(1) Property (Theorem 1.7): Under the same condition ( $k > 2 \max_{P \in \text{Per}(f_\infty)} e_{f_\infty}(P)$ ), the map $f$ satisfies the DML(1) property. Furthermore, every irreducible $f$ -periodic curve is a vertical line.
Existence of Examples (Proposition 1.6): The authors prove that for a general endomorphism $f_\infty$ of $H_\infty \cong \mathbb{P}^{N-1}$ of degree $d$ , the maximum multiplicity is bounded by $(N-1)! \cdot 2^{N-1}$ . Consequently, if the degree gap $k$ is chosen as $(N-1)! \cdot 2^N + 1$ , the condition for the main theorem is satisfied for general $f_\infty$ .
Optimality (Example 1.8): The paper provides examples demonstrating that the condition $k > 2 \max e_{f_\infty}(P)$ is optimal. Specifically, if the inequality is relaxed to $\ge$ , there exist invariant curves that are not vertical lines.
Characterization of Periodic Curves: As a by-product, the authors completely characterize the periodic curves for such maps: they are strictly vertical lines. If the union of these lines is Zariski dense, the lower-degree terms $g_i$ must vanish.

Significance and Claims
The paper claims to advance the understanding of the DML conjecture in higher dimensions by introducing a "local height argument" tailored to polarized systems with a visible degree gap. The significance lies in:

Breaking Isotropy: The condition on the degree gap and ramification is interpreted as a mechanism to "break down the isotropy" of the system, allowing the authors to find two different speeds of local height growth, which is necessary for applying Roth's theorem.
Complete Classification: Unlike previous works that might only establish the DML property, this approach allows for the explicit determination of all periodic curves (identifying them as vertical lines) as a direct consequence of the proof.
Optimality: The authors explicitly address the sharpness of their conditions, showing that the strict inequality is necessary to force the geometric rigidity of the periodic curves.
Limitations in Positive Characteristic: The paper notes that their method relies heavily on Siegel's theorem and Roth's theorem, which do not hold in the same form in positive characteristic. They provide a counterexample in positive characteristic where invariant curves are not vertical lines, suggesting that new methods are required for that setting.

The work is presented as a successor to previous height-based arguments (specifically referencing [XYb]) but adapted to the specific constraints of regular endomorphisms with degree gaps, utilizing the arithmetic function field setting to handle the complex case.