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arxiv: 2604.04881 · v1 · submitted 2026-04-06 · 🧮 math.DS · math.AG· math.NT

Unlikely intersections in families of polynomial skew products

Chatchai Noytaptim , Xiao Zhong This is my paper

Pith reviewed 2026-05-10 19:21 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.NT
keywords polynomial skew productspreperiodic pointsunlikely intersectionsadelic heightspostcritically finitedynamical Manin-Mumfordmoduli space
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The pith

In families of polynomial skew products, two marked points are simultaneously preperiodic for infinitely many parameters if and only if their good heights are equal.

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

The paper classifies the special loci in the moduli space of quadratic regular polynomial skew products that contain a Zariski dense set of postcritically finite points. These loci include families of homogeneous polynomial endomorphisms, split endomorphisms, and maps of the form (x squared, y squared plus b times x) up to conjugacy. It defines a good height for marked points using adelic line bundles and proves that the parameters where two marked points are both preperiodic form an infinite set exactly when the two heights are equal. This height is applied to show that infinite preperiodicity of one point implies the Zariski closure of its orbit lies in a proper subvariety under suitable degree conditions, giving a conditional verification of a relative dynamical Manin-Mumford conjecture.

Core claim

For a family F_t of regular polynomial skew products over a number field K with marked points P_t and Q_t in K[t] x K[t], the set of t0 in the algebraic closure where both P_t0 and Q_t0 are preperiodic for F_t0 is infinite if and only if the good height h of P_t equals that of Q_t. The special loci with dense postcritically finite points are the homogeneous families, the split families, and the (x^2, y^2 + b x) family up to conjugacy.

What carries the argument

The good height h_{P_t} built from adelic line bundles on the quasi-projective variety, which coincides with the canonical height on preperiodic points and controls the accumulation of preperiodic parameters.

If this is right

  • The classification verifies a special case of Zhong's conjecture on special loci.
  • Under degree conditions on P_t, an infinite set of preperiodic parameters forces the forward orbit of P_t to have Zariski closure in a proper subvariety of P^2.
  • Special cases of the DeMarco-Mavraki conjecture are conditionally verified as a relative version of the dynamical Manin-Mumford conjecture.

Where Pith is reading between the lines

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

  • This height criterion could be used to find explicit examples of preperiodic parameters by checking height equality in concrete families.
  • The results suggest that unlikely intersections are controlled by arithmetic invariants like heights in these dynamical settings.
  • Extensions might apply the same method to families in higher dimensions or with more marked points.

Load-bearing premise

The maps are regular polynomial skew products defined over a number field, with the marked points having polynomial coordinates in the parameter and additional unspecified degree conditions for the orbit closure part.

What would settle it

An explicit family of skew products where the good heights of the two marked points differ but infinitely many parameters still make both preperiodic, or where the heights are equal but only finitely many such parameters exist.

read the original abstract

Motivated by the study of unlikely intersection in the moduli space of rational maps, we initiate our investigation on algebraic dynamics for families of regular polynomial skew products in this article. Our goals are threefold. (1) We classify special loci -- which contain a Zariski dense set of postcritically finite points -- in the moduli space of quadratic regular polynomial skew products. More precisely, special loci include families of homogeneous polynomial endomorphisms, families of split endomorphisms, and polynomial endomorphisms of the form $(x^2,y^2+bx)$ up to conjugacy. As a consequence, we verify a special case of a conjecture proposed by Zhong. (2) Let $F_t$ be a family of regular polynomial skew products defined over a number field $K$ and let $P_t, Q_t\in K[t]\times K[t]$ be two initial marked points. We introduce a good height $h_{P_t}(t)$ which is built from the theory of adelic line bundles for quasi projective varieties. We show that the set of parameters $t_0\in \overline{K}$ for which $P_{t_0}$ and $Q_{t_0}$ are simultaneously $F_{t_0}$-preperiodic is infinite if and only if $h_{P_t}=h_{Q_t}$. (3) As an application of $h_{P_t}$, we show that, under some degree conditions of $P_t$, if there is an infinite set of parameters $t_0$ for which the marked point $P_{t_0}$ is preperiodic under $F_{t_0}$, then the Zariski closure of the forward orbit of $P_t$ lives in a proper subvariety of $\mathbb{P}^2$. As a by-product, we conditionally verify a special case of a conjecture of DeMarco--Mavraki which is a relative version of the Dynamical Manin--Mumford Conjecture.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies unlikely intersections in families of regular polynomial skew products on P². It classifies the special loci in the moduli space of quadratic regular polynomial skew products that contain a Zariski-dense set of postcritically finite points: these are (up to conjugacy) families of homogeneous polynomial endomorphisms, split endomorphisms, and maps of the form (x², y² + b x). This verifies a special case of a conjecture of Zhong. For a general family F_t defined over a number field K with marked points P_t, Q_t ∈ K[t] × K[t], the authors introduce a good height h_{P_t} constructed via adelic line bundles on quasi-projective varieties and prove that the set of parameters t₀ ∈ K-bar for which both P_{t₀} and Q_{t₀} are simultaneously F_{t₀}-preperiodic is infinite if and only if h_{P_t} = h_{Q_t}. As an application, under unspecified degree conditions on P_t, an infinite set of preperiodic parameters for P_t implies that the Zariski closure of the forward orbit of P_t lies in a proper subvariety of P²; this conditionally verifies a special case of the DeMarco–Mavraki conjecture (a relative form of the Dynamical Manin–Mumford conjecture).

Significance. If the central claims hold, the work supplies an explicit classification of special loci together with a height-theoretic criterion that detects infinite simultaneous preperiodicity. The adelic-line-bundle construction of the good height is a natural and potentially reusable tool for families on quasi-projective varieties. The conditional verification of the relative Dynamical Manin–Mumford conjecture and the orbit-closure application are concrete advances in higher-dimensional unlikely-intersections problems. The paper therefore strengthens the toolkit for studying post-critically finite parameters and orbit closures in polynomial skew-product families.

major comments (2)
  1. [part (3)] Part (3) of the main results: the statement invokes 'some degree conditions of P_t' without stating them explicitly. Because the conclusion that the Zariski closure of the forward orbit lies in a proper subvariety of P² depends on these conditions, their omission is load-bearing for the application and for the conditional verification of the DeMarco–Mavraki conjecture.
  2. [part (2)] The height-equivalence theorem (part (2)): while the 'only if' direction follows from standard Northcott properties once h_{P_t} is shown to be a good height, the 'if' direction (equal heights imply infinitely many common preperiodic parameters) appears to require that the family F_t lies in one of the special loci classified in part (1). The manuscript should clarify whether the equivalence holds for arbitrary regular skew-product families or only after reduction to the classified cases.
minor comments (2)
  1. The notation h_{P_t}(t) versus h_{P_t} is used interchangeably in the abstract; a uniform notation should be fixed throughout the text.
  2. The abstract refers to 'regular polynomial skew products' without recalling the precise definition (e.g., the condition that the map is regular at infinity); a brief reminder in the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the significance of our results. We address the major comments point by point below, indicating the revisions we will make to improve clarity.

read point-by-point responses

  1. Referee: Part (3) of the main results: the statement invokes 'some degree conditions of P_t' without stating them explicitly. Because the conclusion that the Zariski closure of the forward orbit lies in a proper subvariety of P² depends on these conditions, their omission is load-bearing for the application and for the conditional verification of the DeMarco–Mavraki conjecture.

    Authors: We agree that the degree conditions on P_t were not stated explicitly in the theorem statement of part (3), which affects the readability of the application and the conditional verification of the DeMarco–Mavraki conjecture. In the revised manuscript, we will explicitly state the precise degree conditions on P_t that are used in the proof (specifically, those ensuring the orbit closure lies in a proper subvariety when there are infinitely many preperiodic parameters). We will also add a short explanation of their role in the argument. revision: yes

  2. Referee: The height-equivalence theorem (part (2)): while the 'only if' direction follows from standard Northcott properties once h_{P_t} is shown to be a good height, the 'if' direction (equal heights imply infinitely many common preperiodic parameters) appears to require that the family F_t lies in one of the special loci classified in part (1). The manuscript should clarify whether the equivalence holds for arbitrary regular skew-product families or only after reduction to the classified cases.

    Authors: The equivalence in part (2) holds for arbitrary families of regular polynomial skew products F_t; the 'if' direction does not require reduction to the special loci classified in part (1). The proof proceeds directly from the construction of the good height h_{P_t} via adelic line bundles on the quasi-projective parameter space: when h_{P_t} = h_{Q_t}, the set of parameters t_0 where both P_{t_0} and Q_{t_0} are preperiodic consists of points of bounded height, and the Northcott property then yields infinitely many such parameters over the algebraic closure. Part (1) addresses a distinct question concerning Zariski-dense sets of postcritically finite maps in the moduli space and is not invoked in the proof of the height equivalence. To prevent misinterpretation, we will add a clarifying remark in the introduction and immediately following the statement of the theorem in part (2). revision: partial

Circularity Check

0 steps flagged

Minor self-citation present but derivation remains independent

full rationale

The central results rely on standard adelic height constructions from line bundles on quasi-projective varieties and Zariski topology arguments, which are external and do not reduce to the paper's own fitted quantities or equations. The equivalence of infinite simultaneous preperiodic parameters with equal heights h_Pt = h_Qt follows from Northcott properties without internal redefinition. A minor self-citation occurs when verifying a special case of a conjecture proposed by coauthor Zhong, but this is not load-bearing for the main theorems or height equivalence. No self-definitional, fitted-prediction, or ansatz-smuggling patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The paper relies on standard background from algebraic geometry and arithmetic dynamics; the main addition is the construction of the good height from existing adelic theory rather than new axioms or entities.

axioms (3)
  • standard math Standard properties of heights arising from adelic line bundles on quasi-projective varieties
    Invoked to define the good height h_Pt used in the preperiodicity criterion
  • standard math Zariski topology and the notion of Zariski density in algebraic varieties
    Used to state that special loci contain a Zariski dense set of postcritically finite points
  • domain assumption Basic facts about preperiodic and postcritically finite points for polynomial endomorphisms
    Core background for the study of unlikely intersections in dynamical systems
invented entities (1)
  • good height h_Pt no independent evidence
    purpose: To provide an arithmetic measure of marked points in the family that detects simultaneous preperiodicity
    Constructed from the theory of adelic line bundles; no independent falsifiable prediction outside the paper is stated

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