{"id":"155a35df-8f04-4b16-ac60-870a0e6944f9","arxiv_id":"2412.15141","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Common zeros of two compositionally independent iterated morphisms are non-Zariski-dense for Henon maps, split endomorphisms of (P^1)^n, and regular polynomial skew products over number fields.","lead":"This paper shows that for several families of iterated maps on algebraic varieties, the points where two independent maps hit the same target after iteration form a small, non-dense set. These families include Henon maps, coordinatewise maps on products of projective lines, and polynomial skew products over number fields.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.13's slice-wise detection rests on an unproved claim that the vertical fiber over each periodic base point contains infinitely many preperiodic points; without it, the Tits-alternative step and Theorem 1.6 fail.","rationale":"The reader's weakest_assumption identifies Lemma 4.13 and the slice-wise detection as the most fragile part of the argument. My reading agrees: the proof of Lemma 4.13 has a genuine missing step, namely the unbounded growth of pi_y(S2) when L is enlarged. This step is load-bearing because the entire Tits-alternative Corollary 4.16 and therefore Theorem 1.6 rely on deriving slice-wise preperiodic-set equality from Prep(F1)=Prep(F2). The concern is not a demonstrated counterexample; the missing claim is almost certainly true, since the return map on a vertical fiber over a periodic point has degree d^{n0}>=2 and hence infinitely many preperiodic points. However, the manuscript does not state or prove this, and the proof as written is too compressed to certify the point. A careful reader cannot rule out a hidden exceptional case without the explicit degree computation and Northcott argument. Therefore the appropriate verdict remains CONDITIONAL, exactly as the reader proposed: the result is credible and likely correct, but the proof needs to be expanded at this precise location before it can be accepted without reservation. I find no other concern that is more load-bearing: the equidistribution, local-to-global, and polynomial-hull steps are standard and cited to the literature, and the remaining Ritt-decomposition analysis, though intricate, is internally coherent in the parts I could verify. The proposed concrete test—checking the missing growth argument for the fiber sets—would settle whether the concern lands or is merely an expositional gap.","tokens_in":37588,"tokens_out":29331,"duration_ms":256717,"concrete_test":"Write out the missing verification for Lemma 4.13: fix a periodic point x0 of period n0, and prove that the fiber return map R(y)=G_{1,n0-1,x0}(y) composed with ... composed with G_{1,0,x0}(y) has degree d^{n0}>=2, using the constant nonzero leading coefficient of the y^d term in g1(x,y). Then show that for every N there is a finite extension L_N of the field generated by the coefficients and x0 such that |Prep(R)(L_N)| >= N, by adjoining N distinct preperiodic points of R (which exist because every polynomial of degree >=2 over C has infinitely many preperiodic points). If this succeeds, the slice-wise detection stands; if a periodic x0 is found for which the fiber contains only finitely many F1-preperiodic points, then the conclusion of Lemma 4.13 is false and the proof of Theorem 1.6 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.6 reduces compositional independence of two regular polynomial skew products to Corollary 4.16, whose proof passes through Lemma 4.13. Lemma 4.13 asserts that Prep(F1)=Prep(F2) implies, for every periodic x0 of the common first-coordinate map f, equality of preperiodic sets of the two one-dimensional return maps G_{3,n0-1,x0} and G_{4,n0-1,x0}. The proof defines S1=Prep(F1)(L) for a finitely generated field L containing x0 and all coefficients, lets S2 be the subset with first coordinate x0, and observes that pi_y(S2) is invariant under both return maps. The decisive sentence is: 'if we enlarge the field L, the size of pi_y(S2) will keep increasing.' This is asserted without proof. Without it, one only knows that the two return maps share the finite set pi_y(S2)(L), which is insufficient to apply [BD11, Theorem 1.2] and conclude equality of all preperiodic points. The missing justification is that the return map R = g_{1,n0-1,x0} composed with ... composed with g_{1,0,x0} 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 then forces the finite sets Prep(R)(L) to grow unboundedly as L ranges over finite extensions. This is plausible and likely true, but it is not written; if it failed for some periodic x0 (e.g., if the return map had degree 1 or were exceptional), Lemma 4.13 would collapse, taking Proposition 4.11, Corollary 4.16, and Theorem 1.6 with it. The manuscript should supply the missing degree computation and the Northcott-based growth argument explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":17,"tokens_out":12418,"duration_ms":152782,"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":[{"comment":"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.","section":"Lemma 4.13"},{"comment":"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]).","section":"Appendix A, Proposition 5.1"},{"comment":"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.","section":"Proposition 4.11, Case II"}],"minor_comments":[{"comment":"The notation 'f^{xny}' and the surrounding sentence clarifying the n-th power are confusing; please rewrite this paragraph in standard notation.","section":"Introduction, Question 1.1 paragraph"},{"comment":"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":"Theorem 2.8 and Definition 2.7"},{"comment":"The paper uses both 'PrePer' and 'Prep' for preperiodic points; please make the notation uniform.","section":"Section 5.1"},{"comment":"There are several typos in this appendix ('Nullstellensatz', 'Thers', 'Cpxq', 'contiuity'); these should be corrected in a revision.","section":"Appendix A, proof of Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The central line of proof is coherent and the results, if correct after repair, would be a meaningful advance. The two main concerns are both fixable in a revision: Lemma 4.13 needs a rigorous justification of the field-enlargement step, and Appendix A contains an invalid proof of a fundamental growth estimate. The paper also leans heavily on external one-dimensional results, but the use is clearly cited and not circular. I would recommend major revision rather than rejection, with the expectation that the authors can supply the missing arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.\n\nMy 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.","headline":"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.","tokens_in":38508,"tokens_out":1528,"would_cite":false,"duration_ms":12902,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P05","37P30","37P50"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["arithmetic dynamics","arithmetic equidistribution","compositional independence","free semigroup","polynomial decomposition","polynomial skew products","Hénon type maps","Hsia-Tucker question"],"falsifier":"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.","tokens_in":37338,"feed_emoji":"🌀","tokens_out":9529,"duration_ms":55225,"temperature":0.7,"pith_summary":"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.","feed_headline":"Common zeros of iterated maps are never Zariski dense in three classes","feed_subtitle":"Answering Hsia-Tucker: these map families cannot have Zariski-dense common iterated zeros.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Posed Question 1.1 and proved the one-dimensional polynomial case that this paper extends to three higher-dimensional classes.","marker":"[HT17]"},{"why":"Proved finiteness of common zeros for iterated rational maps on $\\mathbb{P}^1$, the one-dimensional precedent for the equidistribution-and-contradiction strategy.","marker":"[NZ24]"},{"why":"Supplies the arithmetic equidistribution theorem that turns a generic small-height sequence into equality of equilibrium measures at all places.","marker":"[Yu08]"},{"why":"Provides the one-dimensional free-semigroup dichotomy for rational functions used in the induction for endomorphisms of $(\\mathbb{P}^1)^n$ and in the skew-product argument.","marker":"[BHPT24]"},{"why":"Shows that Hénon type maps sharing their periodic points are compositionally dependent, and supplies the convex-hull lemma identifying the filled Julia set with the hull of the measure support.","marker":"[DF17]"},{"why":"Gives the unlikely-intersection principle used to conclude that two one-dimensional maps sharing infinitely many preperiodic points have equal preperiodic sets.","marker":"[BD11]"},{"why":"Constructs local and global canonical heights for affine plane automorphisms, giving the height-zero characterization of periodic points used in the local-to-global step.","marker":"[Ka13]"},{"why":"Provides the linearization and commutation results for polynomial decompositions that control the slice maps in the skew-product free-semigroup dichotomy.","marker":"[Pa20]"},{"why":"Shows polynomials with the same Julia set are built from a common polynomial by linear left and right factors, a key input when slice maps are non-special.","marker":"[SS95]"}],"fun_headline_variants":["No Zariski-dense common zeros for iterated Hénon, splits, and skew products","Three map families: iterated zeros never Zariski dense","Hsia-Tucker resolved: common iterated zeros not Zariski dense in these classes","Hénon, skew products, and split endomorphisms: no Zariski-dense common zeros"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No Zariski-dense common zeros for iterated Hénon, splits, and skew products","Three map families: iterated zeros never Zariski dense","Hsia-Tucker resolved: common iterated zeros not Zariski dense in these classes","Hénon, skew products, and split endomorphisms: no Zariski-dense common zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001031,"raw_usage":{"total_tokens":4323,"prompt_tokens":905,"completion_tokens":3418,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":3324}},"tokens_in":521,"tokens_out":3418,"duration_ms":16061,"temperature":1.0,"reasoning_tokens":3324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:35:15.928057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}