Towards Common Zeros of Iterated Morphisms

Chatchai Noytaptim; Xiao Zhong

REVIEW 3 major objections 4 minor 15 references

Towards Common Zeros of Iterated Morphisms

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that for Hénon type maps, split endomorphisms of $(\mathbb{P}^1)^n$, and regular polynomial skew products, the set of points where some iterates of two compositionally independent maps coincide with a fixed morphism is…

desk verdict Real progress on Hsia-Tucker for three classes, but the skew-product main theorem hinges on an unproved growth claim in Lemma 4.13 that needs to be fixed before the result is fully established. read the letter →

arxiv 2412.15141 v1 pith:WE3X32HM submitted 2024-12-19 math.AG math.DSmath.NT

Chatchai Noytaptim , Xiao Zhong This is my paper · ORCID

classification math.AGmath.DSmath.NT MSC 37P0537P3037P50

keywords arithmeticdynamicsequidistributioncompositionalindependencefreesemigrouppolynomialdecompositionskewproductsHénontypemapsHsia-Tuckerquestion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper takes on Hsia and Tucker's question: if two self-maps of a variety are compositionally independent, can the set of points where some iterate of one equals some iterate of the other and both equal a prescribed morphism be Zariski dense? The authors prove that the answer is no for three classes defined over number fields: Hénon type automorphisms of the affine plane, coordinatewise (split) endomorphisms of $(\mathbb{P}^1)^n$, and regular polynomial skew products on $\mathbb{A}^2$. The proof runs through a common four-step scheme: any hypothetical Zariski dense solution sequence must have canonical height tending to zero; arithmetic equidistribution then forces the two maps to induce the same measures at every place; a local-to-global height argument upgrades this to equality of preperiodic points; and a free-semigroup dichotomy for the relevant map class then forces the two maps to be compositionally dependent, contradicting the hypothesis. A by-product is a dichotomy for semigroups generated by two regular polynomial skew products: either their preperiodic sets coincide, or the semigroup contains a nonabelian free subsemigroup.

What carries the argument

The argument is carried by three interacting objects. First, a canonical height attached to each map, built from $v$-adic Green functions, detects smallness: a sequence solving $F^m=G^n=C$ at growing exponents must have both canonical heights tending to zero. Second, arithmetic equidistribution of small points forces the associated equilibrium measures to coincide at every place; a convex-hull lemma then identifies the filled Julia set with the polynomial hull of the measure support, and a local-to-global principle turns equality of filled Julia sets into equality of the preperiodic point sets. Third, a free-semigroup dichotomy for the map class converts equality of preperiodic sets into compositional dependence: for endomorphisms of $(\mathbb{P}^1)^n$ it is obtained by induction on $n$ from the one-dimensional dichotomy for rational functions, while for regular polynomial skew products it follows from the classical polynomial decomposition theorem, specialization to periodic base points, and detailed analysis of the slice maps $g(x_0,y)$. The final contradiction is that a Zariski dense solution sequence would make $F$ and $G$ compositionally dependent, against the hypothesis.

What would settle it

Construct two compositionally independent regular polynomial skew products over the complex numbers with the same preperiodic set whose composition semigroup contains a nonabelian free subsemigroup; Corollary 4.16 predicts none exists, so such a pair would refute the central mechanism. A cheaper test is to compute, for explicit $F=(f,g_1)$ and $G=(f,g_2)$ with a shared preperiodic set, the slice preperiodic sets at one periodic base point $x_0$ and compare them with the prediction of Lemma 4.13.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.6, is that for regular polynomial skew products $F$ and $G$ of degree at least 2 over a number field $K$ that are compositionally independent, with $C$ any morphism that is not a compositional power of $F$ or $G$, the set of $(x,y)$ in $\mathbb{A}^2_K$ for which $F^m(x,y)=G^n(x,y)=C(x,y)$ for some positive integers $m$ and $n$ is not Zariski dense in $\mathbb{A}^2$. The same conclusion is proved for Hénon type automorphisms of $\mathbb{A}^2$ (Theorem 1.2) and for split endomorphisms of $(\mathbb{P}^1)^n$ whose coordinate maps all have the same degree (Theorem 1.3). The paper thereby gives an affirmative answer to Hsia-Tucker Question 1.1 for these three classes. Along the way it establishes a Tits-alternative-type result for regular polynomial skew products: equality of preperiodic sets implies the composition semigroup contains no nonabelian free subsemigroup, and non-equality implies it does contain one.

Load-bearing premise

The proof depends on the claim that equality of the full preperiodic sets of two skew products can be detected slice by slice: after passing to suitable elements of the semigroup, equality above every periodic base point of the common base map forces the whole semigroup to be non-free; if that slice-wise test can fail, the final contradiction collapses.

Editorial extensions

If this is right

Hsia-Tucker Question 1.1 has an affirmative answer for Hénon type automorphisms of $\mathbb{A}^2$ over number fields.
Hsia-Tucker Question 1.1 has an affirmative answer for split endomorphisms of $(\mathbb{P}^1)^n$ over number fields, extending the one-dimensional rational-map case.
Hsia-Tucker Question 1.1 has an affirmative answer for regular polynomial skew products on $\mathbb{A}^2$ over number fields.
For two regular polynomial skew products, equality of their preperiodic point sets is equivalent to the absence of a nonabelian free subsemigroup in the composition semigroup they generate.
In all three classes, the non-density conclusion allows the two iterate exponents to vary independently, not just to be equal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same four-step scheme should extend to regular polynomial endomorphisms of $\mathbb{A}^k$ that preserve a fibration over a lower-dimensional base, wherever a slice-wise free-semigroup dichotomy can be established.
The dichotomy for skew products suggests a general principle for dominant endomorphisms: sharing the full preperiodic set is an extremely rigid condition that forces algebraic relations among the maps, not merely dynamical coincidence.
A quantitative refinement would be to bound the height of the finitely many exceptional solutions when $C$ is not compositionally related to $F$ or $G$, or to show that such solutions lie on a specific low-degree curve determined by $F$, $G$, and $C$.
The method also implies that if $C$ is chosen generically, the equation $F^m=G^n=C$ has only finitely many solutions total, which can be tested numerically for explicit polynomial skew products.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read this paper with some care, and I think the honest summary is: genuinely new results, a mostly sound strategy, and one load-bearing gap in the skew-product section that is currently asserted rather than proved. The Hénon case is largely a repackaging of Dujardin-Favre with height machinery, but the Tits-alternative-type dichotomy for regular polynomial skew products (Theorem 1.7 / Corollary 4.16) is a real new contribution, and the split-endomorphism case is a clean induction. The paper is worth engaging with.

The soft spot is Lemma 4.13. The proof needs the claim that, after enlarging the finitely generated field L, the projection pi_y(S2) keeps growing in size. That is precisely the step that would let the authors go from a finite common invariant set to infinitely many shared preperiodic points, and then apply [BD11, Theorem 1.2]. The manuscript simply says 'the size of pi_y(S2) will keep increasing' with no argument. The stress-test note is right: this is not a cosmetic omission, because without it the two return maps only share a finite set over L, which is insufficient for the concluding equality of preperiodic sets. A fix should be possible: the return map is a polynomial of degree d^{n0} >= 2, hence has infinitely many preperiodic points over the algebraic closure, and Northcott finiteness over finitely generated fields should force the finite sets to grow under field extension. But that argument has to be written down, and it has to handle the exceptional cases where the return map might have degree 1 or be special. The same applies to the proof sketches for Theorem 2.11 and Proposition 5.8: they are likely fine, but they leave real work to the reader.

I do not think there is a hidden circular dependency. The self-citation [NZ24] is used only for the n=1 base case, which is legitimate. The external inputs (BHPT24, DF17, Ritt decomposition, Schmidt-Steinmetz) are standard for this kind of argument.

My recommendation: send this to a serious referee. The paper is not ready for acceptance as-is, but the main ideas are credible and the gap in Lemma 4.13 looks fixable. A referee should ask for a complete proof of Lemma 4.13, including the field-growth step and a discussion of exceptional periodic points, plus expansions of the two proof sketches. If the authors can supply those, this would be a solid contribution to arithmetic dynamics.

Referee Report

3 major / 4 minor

Summary. The paper addresses a question of Hsia and Tucker on the Zariski non-density of common solutions to F^m(x) = G^n(x) = C(x) for compositionally independent dominant morphisms F,G. It gives affirmative answers for Hénon-type polynomial automorphisms of A^2, coordinatewise endomorphisms of (P^1)^n, and regular polynomial skew products of A^2, all defined over number fields. The strategy is uniform: a hypothetical Zariski-dense sequence of common solutions is shown to be dynamically small for both maps, arithmetic equidistribution forces equality of the associated equilibrium measures, a local-to-global principle then yields equality of preperiodic sets, and a Tits-alternative-type theorem converts equality of preperiodic sets into compositional dependence, giving a contradiction. A by-product is a Tits alternative for semigroups generated by two regular polynomial skew products (Corollary 4.16 and Remark 4.17).

Significance. If the proofs are completed as indicated, the results constitute substantial progress on Question 1.1 in genuinely higher-dimensional settings. The paper introduces and uses canonical heights and local Green functions for polynomial skew products, establishes local-to-global principles for preperiodic points, and supplies the missing algebraic step (shared preperiodic sets imply non-freeness) for this class. The reliance on external machinery ([BHPT24], [BD11], [DF17], [SS95]) is clearly documented, and there are no fitted parameters or ad-hoc normalizations. The three main theorems are falsifiable statements about Zariski non-density, and the by-product Tits alternative is of independent interest. However, the manuscript currently contains proof gaps and an incorrect argument in a foundational appendix, so the significance is contingent on repair.

major comments (3)

[Lemma 4.13] The proof asserts, without justification, that 'if we enlarge the field L, the size of π_y(S_2) will keep increasing.' This is load-bearing: the argument only shows that the fixed finite set π_y(S_2)(L) is invariant under the two return maps, which is insufficient to apply [BD11, Theorem 1.2]. One needs to prove that for each periodic x_0 the return map R = G_{3,n_0-1,x_0} ∘ ⋯ ∘ G_{3,0,x_0} has degree d^{n_0} ≥ 2 and hence has infinitely many preperiodic points over the algebraic closure, and that arbitrarily many of the corresponding points (x_0,y) lie in Prep(F_1)(L) as L ranges over finitely generated extensions. This is plausible and probably true, but it is not written. Please expand this step so that the infinite-intersection conclusion in Lemma 4.13 is rigorously justified; without it, Corollary 4.16 and Theorem 1.6 lose their main algebraic input.
[Appendix A, Proposition 5.1] The proof of the lower bound in inequality (5.1) claims that for p(x) ∈ C[x] there exists e ≥ 0 with x^e in the ideal generated by p(x). This is false in general (e.g., p(x) = x+1 divides no monomial x^e). Consequently, the subsequent derivation of C'_v max(|x|,|y|)^d ≤ max(|p(x)|,|q(x,y)|) does not go through. In fact, the inequality as stated is false at points where both p and q vanish, e.g., at a root of p with y = 0 and max(|x|,|y|) = 1. Since Corollary 5.2 and the height construction in §5 rely on this growth estimate, the proposition should be replaced by a correct statement (or a correct proof, possibly using the projective lift F and the fact that a regular skew product extends to an endomorphism of P^2, or by citing the standard result from [DFR23]).
[Proposition 4.11, Case II] The step 'by a series of detailed studies on the Julia set of polynomials [Bea90], [Bea92], [BE87], [SS95], we have G_1,x_0 = σ ∘ G_2,x_0' is extremely compressed. Since Proposition 4.11 is used in Corollary 4.16 and hence in Theorem 1.6, this should be spelled out or explicitly reduced to a named theorem with the exact hypotheses checked, in particular that the equal preperiodic sets imply equality of Julia sets and that the relevant polynomials are non-special.

minor comments (4)

[Introduction, Question 1.1 paragraph] The notation 'f^{xny}' and the surrounding sentence clarifying the n-th power are confusing; please rewrite this paragraph in standard notation.
[Theorem 2.8 and Definition 2.7] The notation '~hf pP2q' in the proof of Theorem 2.8 is not formally defined; the global height of the projective space with respect to the adelic metric should be defined or replaced by a clearer inequality.
[Section 5.1] The paper uses both 'PrePer' and 'Prep' for preperiodic points; please make the notation uniform.
[Appendix A, proof of Proposition 5.1] There are several typos in this appendix ('Nullstellensatz', 'Thers', 'Cpxq', 'contiuity'); these should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation reduces to external one-dimensional theorems and height equidistribution, not to the paper's own inputs.

full rationale

This paper's derivation chain is self-contained in the relevant sense. Theorem 1.6 is proved by contradiction from the small-height Proposition 5.8, arithmetic equidistribution (Theorem 5.10), the local-to-global Lemma 5.6, and Corollary 4.16. Corollary 4.16 is in turn a coordinate analysis of regular skew products whose key inputs are the one-dimensional Tits alternative [BHPT24, Corollary 4.11], the unlikely-intersection theorem of Baker-DeMarco [BD11, Theorem 1.2], classical results on polynomials with identical Julia sets [SS95], and Ritt decomposition theory. None of these is the paper's own conclusion. The only self-citation, [NZ24], is used historically in the Introduction and as a remark that the n=1 case of Theorem 1.3 was previously known; it does not appear in the proof of Theorem 1.6, Lemma 4.13, or Corollary 4.16. The most delicate step, Lemma 4.13, asserts that equality of the global preperiodic sets forces equality of return-map preperiodic sets slice-by-slice; the contested sentence 'if we enlarge the field L, the size of pi_y(S2) will keep increasing' is an unexpanded Northcott-finiteness argument, not a definitional identification of the target with the input. No fitted parameters are renamed as predictions, no ansatz is smuggled in through a self-citation, and no uniqueness theorem is imported from the authors' prior work. Hence no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims introduce no fitted constants, no ad hoc normalizations, and no new speculative objects. The proof imports external results from arithmetic equidistribution, Ritt decomposition, and one-dimensional dynamics; these are standard or published, though some are deep.

assumptions (7)

standard math Yuan's arithmetic equidistribution theorem for points of small height with semipositive adelic metrics.
Invoked in Theorem 2.8 and Theorem 5.10 to pass from smallness of a generic sequence to equality of equilibrium measures.
standard math Baker-DeMarco unlikely intersection theorem [BD11, Theorem 1.2].
Used in Lemma 3.3 and Lemma 4.13 to conclude equality of preperiodic sets from infinitely many shared preperiodic points.
domain assumption Tits alternative for rational functions on P^1 [BHPT24, Corollary 4.11 and Theorem 1.3].
Used to show non-freeness of first-coordinate semigroups and to produce free subsemigroups when preperiodic sets differ.
standard math Ritt's polynomial decomposition theorem (Theorem 4.8).
Central tool in Section 4 for analyzing decompositions of iterates of polynomial skew products.
domain assumption Results on polynomials with identical Julia sets [SS95, Bea90, Bea92, BE87].
Used in Proposition 4.11 and Lemma 4.12 to relate maps sharing the same preperiodic sets to common polynomial cores.
domain assumption Kawaguchi's canonical height and Green function theory for Henon maps [Ka13].
Provides the small-point criterion, local-to-global properties, and semipositive adelic metric structure used in Section 2.
standard math Northcott property for heights over number fields and finitely generated fields [Mor00].
Used to make preperiodic sets finite in certain slices and to detect preperiodicity from vanishing of canonical height.

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Pith. "Pith review of Towards Common Zeros of Iterated Morphisms." pith.science (2026). https://pith.science/paper/WE3X32HM

@misc{pith2026241215141,
  author       = {Pith},
  title        = {Pith review of: Towards Common Zeros of Iterated Morphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WE3X32HM}},
  note         = {Machine review of arXiv:2412.15141}
}

abstract

Recently, the authors have proved the finiteness of common zeros of two iterated rational maps under some compositional independence assumptions. In this article, we advance towards a question of Hsia and Tucker on a Zariski non-density of common zeros of iterated morphisms on a variety. More precisely, we provide an affirmative answer in the case of H\'{e}non type maps on $\mathbb{A}^2$, endomorphisms on $(\mathbb{P}^1)^n$, and polynomial skew products on $\mathbb{A}^2$ defined over $\overline{\mathbb{Q}}$. As a by-product, we prove a Tits' alternative analogy for semigroups generated by two regular polynomial skew products.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Astorg and F

[AB23] M. Astorg and F. Bianchi. Hyperbolicity and bifurcations in holomorphic fam- ilies of polynomial skew products . Amer. J. Math. 145 (2023), no. 3, 861–898. [ABD` 16] M. Astorg et al.. A two-dimensional polynomial mapping with a wandering Fatou component. Ann. of Math. (2) 184 (2016), no. 1, 263–313. [AR04] N. Ailon and Z. Rudnick. Torsion points on...

work page 2023

[2] [2]

113 (2004), no.1, 31–38

Acta Arithmetica. 113 (2004), no.1, 31–38. [BCZ03] Y. Bugeaud, P. Corvaja, and U. Zannier. An upper bound for the G.C.D. of an ´ 1 and bn ´

work page 2004

[3] [3]

243 (2003), no

Mathematische Zeitschrift. 243 (2003), no. 1, 79–84. [BD11] M. Baker and L. DeMarco. Preperiodic points and unlikely intersections. Duke Mathematical Journal. 159 (2011), no. 1, 1–29. [Bea90] A. F. Beardon. Symmetries of Julia sets . Bull. London Math. Soc. 22 (1990), no. 6, 576–582. [Bea92] A. F. Beardon. Polynomials with identical Julia sets . Complex V...

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[4] [9]

Noytaptim and X

[NZ24] C. Noytaptim and X. Zhong. A ﬁniteness result for common zeros of iterates of rational maps . Preprint arXiv:2405.15104 (2024). [Os16] A. Ostafe. On some extensions of the Ailon-Rudnick theorem . Monatshefte f¨ ur Mathematik.181 (2016), no. 2, 451–471. [Pa20] F. Pakovich. Finiteness theorems for commuting and semiconjugate ratio nal functions. Conf...

arXiv 2024

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Friedland and J

[FM89] S. Friedland and J. Milnor. Dynamical properties of plane polynomial auto- morphisms. Ergodic Theory and Dynamical Systems 9 (1989), no. 1, 67–99. [FRL06] C. Favre and J. Rivera-Letelier. ´Equidistribution quantitative des points de petite hauteur sur la droite projective . Math. Ann. 335 (2006), no. 2, 311–

work page 1989

[6] [361]

[FS99] J. E. Fornæss and N. Sibony. Complex dynamics in higher dimension . In: everal complex variables (Berkeley, CA, 1995–1996). Math. Sci. Res. Inst. Publ., Vol

work page 1995

[7] [1983]

[Le13] C. G. Lee. The equidistribution of small points for strongly regular p airs of polynomial maps. Mathematische Zeitschrift. 275 (2013), no. 3–4, 1047–1072. [Mil06] J. Milnor. Dynamics in one complex variable . Third. 160 (2006), Annals of Mathematics Studies. Princeton University Press, Princet on, NJ, viii+304. 42 CHATCHAI NOYTAPTIM AND XIAO ZHONG ...

work page 2013

[8] [1986]

[In14] P. Ingram. Canonical heights for H´ enon maps . Proceedings of the London Mathematical Society. Third Series. 108 (2014), no. 3, 780–808. [Ji20] Z. Ji, Non-wandering Fatou components for strongly attracting po lynomial skew products. J. Geom. Anal. 30 (2020), no. 1, 124–152. [Ji23] Z. Ji, Non-uniform hyperbolicity in polynomial skew products . Int....

work page 2014