Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic

Junyi Xie; She Yang

arxiv: 2504.02295 · v2 · submitted 2025-04-03 · 🧮 math.DS · math.AG· math.NT

Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic

Junyi Xie , She Yang This is my paper

Pith reviewed 2026-05-22 22:15 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.NT

keywords dynamical Mordell-Lang problempositive characteristicautomorphisms of surfacesprojective surfacesarithmetic dynamicsorbit intersections

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The pith

The dynamical Mordell-Lang problem holds for automorphisms of projective surfaces in positive characteristic.

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

The paper establishes that the dynamical Mordell-Lang problem admits a solution when the map is an automorphism of a projective surface over a field of positive characteristic. This resolves the question of how orbits intersect subvarieties in a structured way for this class of systems. A sympathetic reader cares because the result supplies a positive-characteristic version of statements previously known in characteristic zero, supporting uniform statements about orbit behavior independent of the base field characteristic.

Core claim

The authors solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces, establishing the expected algebraic structure on the set of points whose forward orbits meet a given subvariety in infinitely many points.

What carries the argument

The dynamical Mordell-Lang problem, which asks for a description of points with infinite orbit-subvariety intersections under iteration of the automorphism.

If this is right

The intersections of orbits with subvarieties admit a finite union decomposition into periodic components and structured cosets.
The result applies to every projective surface, not merely special classes such as rational or K3 surfaces.
Statements about orbit finiteness or density can now be made uniformly across all characteristics for this setting.

Where Pith is reading between the lines

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

This surface case may serve as a stepping stone toward the problem for threefolds or for non-invertible maps.
Positive characteristic introduces no extra obstructions beyond those already present in characteristic zero for surface automorphisms.
Similar techniques could be tested on other classes of varieties where the automorphism group is large.

Load-bearing premise

The maps considered are automorphisms, which are invertible, rather than arbitrary endomorphisms of the surfaces.

What would settle it

An explicit automorphism of a projective surface over a field of positive characteristic together with a subvariety whose set of points with infinite orbit intersections fails to be a finite union of periodic orbits and cosets of algebraic subgroups would falsify the claimed solution.

read the original abstract

We solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces.

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

Xie and Yang claim a solution to the dynamical Mordell-Lang problem for automorphisms of projective surfaces in positive characteristic, but the supplied text contains only the claim with no argument or method shown.

read the letter

The paper states that it solves the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces. That is the central announcement to note first. Extending such results beyond characteristic zero is not automatic, given the extra complications that arise with inseparability and orbit behavior in char p, so a working argument here would fill a clear gap in the literature on arithmetic dynamics. The scope is narrow and explicit: only automorphisms (invertible maps) and only surfaces, which probably simplifies the orbit-intersection questions enough to make progress possible. That restriction itself is a reasonable choice and shows the authors are targeting a case where the statement can be made precise. The abstract is direct about what is being claimed, which is better than vague language. Beyond that, the text gives nothing else. There is no sketch of the argument, no key lemma, no reduction step, and no indication of how positive-characteristic difficulties are handled. Without any of that visible, it is not possible to check whether the mathematics actually supports the claim or whether some assumption about the surface case or the automorphism property is doing hidden work. The citation pattern and any use of prior results in char p also cannot be evaluated. This version is mainly of interest to people already following the dynamical Mordell-Lang conjecture and its extensions to positive characteristic. A specialist might want to see the full paper to check the details, but only if the proof is actually there. As presented, the work does not supply enough grounding to merit sending it to referees. The claim is stated cleanly, but the supporting evidence is absent from what we have.

Referee Report

1 major / 0 minor

Summary. The manuscript consists solely of a title and the one-sentence abstract claiming to solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces. No proof outline, reduction steps, invocation of prior results, or technical arguments appear in the supplied text.

Significance. A correct solution to the dynamical Mordell-Lang problem for this class of maps would constitute a substantial advance in arithmetic dynamics over fields of positive characteristic, where inseparability and orbit-density phenomena differ from characteristic zero. The manuscript supplies no evidence that such a solution has been obtained.

major comments (1)

The manuscript contains no sections, equations, or arguments whatsoever. The central claim that the dynamical Mordell-Lang problem is solved therefore cannot be verified or falsified on the basis of the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We acknowledge that the text supplied for review consists only of the title and abstract, which prevents verification of the central claim.

read point-by-point responses

Referee: The manuscript contains no sections, equations, or arguments whatsoever. The central claim that the dynamical Mordell-Lang problem is solved therefore cannot be verified or falsified on the basis of the provided text.

Authors: We agree that the supplied text lacks any proof, reduction steps, or technical arguments, making the claim unverifiable from what was provided. This appears to be an error in the submission; the complete manuscript containing the solution is on arXiv:2504.02295. We will revise the submission to include the full text and proof. revision: yes

Circularity Check

0 steps flagged

No derivation chain or equations supplied; honest non-finding

full rationale

The provided document contains only the title and a single-sentence abstract stating that the dynamical Mordell-Lang problem is solved for automorphisms of projective surfaces in positive characteristic. No proof outline, equations, parameter fits, self-citations, uniqueness theorems, or ansatzes appear. Without any load-bearing steps visible in the text, none of the enumerated circularity patterns can be exhibited by direct quotation and reduction. This is the expected outcome when a paper's central claim is presented without an inspectable derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in any proof.

pith-pipeline@v0.9.0 · 5525 in / 1024 out tokens · 101070 ms · 2026-05-22T22:15:18.604983+00:00 · methodology

Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)