Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve

Aoyang Zheng; Junyi Xie; She Yang

arxiv: 2602.08608 · v2 · submitted 2026-02-09 · 🧮 math.DS · math.AG· math.NT

Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve

Junyi Xie , She Yang , Aoyang Zheng This is my paper

Pith reviewed 2026-05-16 05:52 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.NT

keywords dynamical Mordell-Lang conjecturesplit self-mapsaffine curveprojective curveendomorphismsalgebraic dynamicsorbitssubvarieties

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The pith

The dynamical Mordell-Lang conjecture holds for split self-maps on the product of an affine curve and a projective curve over the algebraic closure of the rationals.

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

The paper proves the dynamical Mordell-Lang conjecture for split self-maps, where the map is the product of an endomorphism on an affine curve and one on a projective curve, all over the algebraic closure of the rationals. This case mixes affine and projective geometry in a product setting. A sympathetic reader would care because it confirms the expected behavior of orbits intersecting subvarieties in this specific but nontrivial class of dynamical systems on algebraic varieties. The result adds to the growing list of cases where the conjecture is known to hold, which could guide future generalizations to other mixed or split maps.

Core claim

The authors prove that the dynamical Mordell-Lang conjecture is true for split self-maps of the form f = f1 × f2 acting on X × Y, where X is an affine curve and Y is a projective curve over the algebraic closure of the rationals. This means that for any point x and subvariety V, the set of iterates n where f^n(x) lies in V is a finite union of arithmetic progressions unless there is a structural reason for it to be infinite.

What carries the argument

The split self-map, defined as the product of independent endomorphisms acting separately on the affine curve factor and the projective curve factor.

If this is right

Orbit intersections with subvarieties follow the pattern predicted by the conjecture for these product varieties.
The result covers the mixed case of one affine factor and one projective factor.
The proof works over the algebraic closure of the rationals and uses tools available in characteristic zero.
It confirms the conjecture for all split maps in this low-dimensional product setting.

Where Pith is reading between the lines

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

The methods may adapt to products of several curves provided the map remains split.
This leaves open whether similar statements hold when the map does not split into independent factors.
Explicit numerical checks on concrete examples such as the multiplicative group times an elliptic curve could test the sharpness of the result.
If the techniques extend, they might address dynamical questions on surfaces that decompose as curve products.

Load-bearing premise

The self-map must factor as a product of independent endomorphisms on each component of the variety.

What would settle it

An explicit split self-map on an affine curve times a projective curve over the algebraic closure of the rationals together with a point and subvariety where the orbit intersection set is infinite yet not a union of arithmetic progressions.

read the original abstract

We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$.

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, Yang, and Zheng prove the dynamical Mordell-Lang conjecture for split endomorphisms on an affine curve times a projective curve over Q-bar.

read the letter

Xie, Yang, and Zheng have shown that the dynamical Mordell-Lang conjecture holds for split self-maps on the product of an affine curve and a projective curve over the algebraic closure of the rationals. This is the key takeaway from the paper. They focus on the split case, where the map is a product of independent endomorphisms on each factor. This structure allows them to analyze the orbits separately and then combine the information using the geometry of the product to determine when the joint orbit intersects a subvariety. The result is stated cleanly for curves, which keeps the technical details manageable. The paper does a good job of targeting a specific untreated combination: affine by projective under split maps. Earlier results covered more symmetric cases or single varieties, so this fills a small gap without overclaiming generality. The base field being Q-bar is standard and avoids extra complications. The main soft spot is the restriction to split maps and low dimensions. The conjecture remains open for non-split maps or higher-dimensional varieties, and this work does not address those. Since I only have the abstract and the stress-test note, I cannot verify the proof steps in detail, but the description gives no sign of circularity or unstated assumptions that would break the argument. This is for specialists in arithmetic dynamics and Diophantine geometry who track progress on the dynamical Mordell-Lang conjecture. A reader interested in verified cases for mixed affine-projective settings will find it useful. It deserves a serious referee because it provides a concrete proof in a natural setting, even if the advance is incremental. I would recommend sending it to peer review.

Referee Report

0 major / 1 minor

Summary. The manuscript proves the dynamical Mordell-Lang conjecture for split self-maps given by the product of an endomorphism of an affine curve and an endomorphism of a projective curve over the algebraic closure of the rationals. The split structure permits separate analysis of the two orbits, with the product geometry controlling simultaneous returns to a subvariety.

Significance. If the proof holds, the result confirms the conjecture in a mixed affine-projective setting under the split assumption. This provides a concrete, parameter-free instance that may inform extensions to non-split or higher-dimensional cases, relying on standard algebraic-geometry tools without ad-hoc axioms or unproven auxiliaries.

minor comments (1)

[Abstract] The abstract could briefly indicate the main reduction step (separate orbit analysis on each factor) to help readers quickly assess the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The summary accurately captures the main result on the dynamical Mordell-Lang conjecture for split self-maps of an affine curve times a projective curve over the algebraic closure of Q. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper proves the dynamical Mordell-Lang conjecture specifically for split self-maps (f,g) on the product of an affine curve and a projective curve over the algebraic closure of Q. The split structure permits separate orbit analysis on each factor, with the product geometry controlling returns to subvarieties. The abstract and description show no self-definitional reductions, no fitted parameters renamed as predictions, and no load-bearing self-citations that collapse the central claim to its own inputs. The argument rests on external algebraic geometry results and is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof is expected to rely on standard background theorems in algebraic geometry and number theory over number fields; no free parameters or new entities are visible from the abstract.

axioms (1)

standard math Standard results on endomorphisms of curves and their dynamics over algebraically closed fields of characteristic zero.
Invoked implicitly by the statement of the conjecture and its proof.

pith-pipeline@v0.9.0 · 5310 in / 1119 out tokens · 18555 ms · 2026-05-16T05:52:30.925166+00:00 · methodology

Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve

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)