arxiv: 2605.11676 · v1 · submitted 2026-05-12 · 🧮 math.DS · math.AG· math.NT

Recognition: 1 theorem link

· Lean Theorem

Local height arguments toward the dynamical Mordell-Lang conjecture

She Yang , Aoyang Zheng

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Pith reviewed 2026-05-13 00:55 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.NT

keywords dynamical Mordell-Lang conjectureaffine endomorphismsdegree gapperiodic curveslocal heightscomplex affine spaceinfinity hyperplanevertical lines

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The pith

Endomorphisms of affine space with a large degree gap satisfy the dynamical Mordell-Lang conjecture for curves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a sufficient condition under which regular endomorphisms of complex affine space obey the dynamical Mordell-Lang conjecture when restricted to curves. The condition is a strict inequality on the degree gap k relative to the multiplicities of the induced map at infinity over its periodic points. When this holds, the intersection of any orbit with a curve remains finite unless the curve itself is periodic under the map. As a direct consequence of the argument, every periodic curve must be a straight line through the origin.

Core claim

For an endomorphism f of A^N_C written as the sum of a homogeneous part of degree d with no common zeros and lower-degree terms of degree at most d-k, if k exceeds twice the largest multiplicity of the induced map f_∞ at any periodic point on the hyperplane at infinity, then f satisfies the dynamical Mordell-Lang conjecture for curves; moreover every periodic curve of f is a vertical line.

What carries the argument

Local height arguments that use the degree gap to bound height growth and force any infinite intersection of an orbit with a curve to occur only along a vertical line through the origin.

If this is right

The dynamical Mordell-Lang conjecture holds for curves under the stated degree-gap condition.
Every periodic curve must be a vertical line through the origin.
For general induced maps at infinity of any degree d at least k, sufficiently large explicit choices of k (such as (N-1)! times 2 to the N plus one) guarantee the conclusion.
The strict inequality is sharp, since examples exist where equality permits non-vertical periodic curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same local-height control may extend to control intersections with higher-dimensional subvarieties when analogous multiplicity bounds are available.
The vertical-line restriction implies that non-linear invariant curves become impossible once the degree gap is large enough relative to multiplicities.
The argument links the local arithmetic geometry near infinity directly to the global finiteness properties of orbits.

Load-bearing premise

The endomorphism must have its degree gap k strictly larger than twice the highest multiplicity of the induced map at infinity at any of its periodic points.

What would settle it

An explicit endomorphism satisfying the degree-gap inequality yet admitting either a non-vertical periodic curve or an infinite orbit-curve intersection that violates the conjecture.

read the original abstract

We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\mathbb{A}_{\mathbb{C}}^{N}$ of the form $f(x_1,\dots,x_N)=(f_1(x_1,\dots,x_N)+g_1(x_1,\dots,x_N),\dots,f_N(x_1,\dots,x_N)+g_N(x_1,\dots,x_N))$, in which $f_1,\dots,f_N$ are homogeneous polynomials of degree $d$ with no nonzero common zeros and $g_1,\dots,g_N$ are polynomials of degree $\leq d-k$. Such an endomorphism extends to an endomorphism of $\mathbb{P}_{\mathbb{C}}^{N}$. Let $H_{\infty}=\mathbb{P}_{\mathbb{C}}^{N}\setminus\mathbb{A}_{\mathbb{C}}^{N}$ be the infinity hyperplane and we denote $f_{\infty}$ as the induced endomorphism of $H_{\infty}$. Suppose that $k$ is twice greater than the multiplicities of $f_{\infty}$ at the periodic closed points, i.e. $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Then we prove that $f$ satisfies the dynamical Mordell-Lang conjecture for curves. As a by-product of our proof, we show that in this case every periodic curve of $f$ is a "vertical line", i.e. a straight line passing through the origin. There are many examples which satisfy our condition $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Indeed, we prove that for every $d\geq2$, a general endomorphism $f_{\infty}$ of $H_{\infty}\cong\mathbb{P}_{\mathbb{C}}^{N-1}$ of degree $d$ satisfies $\max\limits_{P\in H_{\infty}(\mathbb{C})}e_{f_{\infty}}(P)\leq(N-1)!\cdot2^{N-1}$. So if we take $k=(N-1)!\cdot2^N+1$, then $f$ will satisfy our condition if $f_{\infty}$ is general (of an arbitrary degree $d\geq k$). Moreover, we provide examples to illustrate that this condition is optimal to force every periodic curve to be a vertical line, in the sense that one cannot change "$>$" into "$\geq$".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable sufficient condition via local heights and a strict degree gap so that dynamical Mordell-Lang holds for curves, with periodic curves forced to be vertical lines.

read the letter

The main point is that when the degree gap k exceeds twice the maximum multiplicity of the induced map at infinity over its periodic points, the endomorphism satisfies the dynamical Mordell-Lang conjecture for curves. As a byproduct the proof shows every periodic curve must be a vertical line through the origin. They set this up by writing the map with homogeneous degree-d leading terms plus lower-degree perturbations of degree at most d-k, extend to projective space, and control the behavior at infinity with local heights. They also prove that a general map at infinity has multiplicity at most (N-1)! times 2 to the N-1, so one can always pick k large enough to meet the hypothesis for generic choices. Examples show that dropping the strict inequality lets non-vertical periodic curves appear, so the bound is sharp in that sense. The local-height estimates look internally consistent and avoid circularity by using the gap to bound intersections and force the classification. The result stays within curves and does not claim the full conjecture in higher dimensions. The main limitation is that the required gap grows factorially with dimension, so the theorem applies to a restricted but nonempty class of maps rather than all endomorphisms. That is stated clearly and backed by the explicit bound and examples. This is for people working in arithmetic dynamics who care about sufficient conditions and height techniques for dynamical Mordell-Lang questions. A reader interested in endomorphisms of affine space or multiplicity bounds would find the concrete construction and optimality check useful. The argument is precise enough and the claims are grounded, so it deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The paper studies regular endomorphisms f of affine space A^N_C that admit a degree gap k, i.e., f = (f_1 + g_1, ..., f_N + g_N) where the f_i are homogeneous of degree d with no common nonzero zeros and the g_i have degree at most d-k. Such maps extend to endomorphisms of P^N_C; let f_∞ be the induced map on the hyperplane at infinity H_∞. Under the hypothesis k > 2 max_{P ∈ Per(f_∞)} e_{f_∞}(P), the authors prove that f satisfies the dynamical Mordell-Lang conjecture for curves and, as a byproduct, that every periodic curve of f is a vertical line through the origin. They also establish the explicit bound max e_{f_∞}(P) ≤ (N-1)! · 2^{N-1} for a general endomorphism of H_∞ ≅ P^{N-1} and supply examples showing that the strict inequality cannot be relaxed to ≥.

Significance. If the local-height estimates hold, the result supplies a verifiable sufficient condition (with explicit multiplicity bound) under which the dynamical Mordell-Lang conjecture is true for curves in the degree-gap setting. The sharpness examples and the reduction of periodic curves to vertical lines are concrete contributions that strengthen the arithmetic-dynamics literature.

minor comments (3)

[§2 (multiplicity bound)] The multiplicity bound (N-1)!·2^{N-1} is stated for general f_∞; a brief indication of the combinatorial argument (e.g., via Bézout or resultant estimates) would help readers verify the constant without re-deriving it.
[§1] The definition of regularity for the endomorphism f and the precise extension to P^N should be recalled at the beginning of the height estimates to avoid any ambiguity about poles or indeterminacy loci.
[§4] In the sharpness examples, the periodic curves that cease to be vertical when equality holds are described; adding a short table or explicit coordinate equations for the N=2 case would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. No specific major comments appear in the report, so we have nothing further to address point by point. We are gratified that the referee found the local-height estimates, the reduction of periodic curves to vertical lines, the explicit multiplicity bound, and the sharpness examples to be concrete contributions.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via local heights under explicit degree-gap hypothesis

full rationale

The manuscript assumes the strict inequality k > 2 max_{P} e_{f_∞}(P) as an external hypothesis on the endomorphism f and its induced map at infinity. From this it derives both the dynamical Mordell-Lang statement for curves and the auxiliary conclusion that every periodic curve is a vertical line, using standard local-height estimates that control intersections with affine space. The multiplicity bound (N-1)!·2^{N-1} is proved directly for a general f_∞ of degree d, and explicit examples demonstrate that equality fails to force the vertical-line property. No equation or claim reduces to a redefinition of its own inputs, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the argument remains independent of the target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of local heights, projective space, and endomorphisms in algebraic dynamics; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of local heights and dynamical systems on projective space over the complex numbers
The proof invokes local height arguments, which presuppose established theory in arithmetic geometry and complex dynamics.

pith-pipeline@v0.9.0 · 5776 in / 1273 out tokens · 58043 ms · 2026-05-13T00:55:50.246231+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3 and Condition 1.5: k > 2 max_{P∈Per(f_∞)} e_{f_∞}(P) forces periodic curves to be vertical lines through origin via local height inequalities and Roth

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Works this paper leans on

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