{"id":"e6aafb10-26d3-41a9-9d99-1c0de7446c04","arxiv_id":"2607.19252","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Any two complex polynomials either share all preperiodic points or have a uniformly bounded number of common preperiodic points, with the bound depending only on the degrees.","lead":"A new proof shows that over non-Archimedean fields, two polynomials with the same Julia set must be dynamically related as long as their dynamics is not tame. The authors then use this to prove a uniform bound on common preperiodic points of complex polynomials, resolving the polynomial case of a conjecture of DeMarco–Krieger–Ye.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.6 relies on an unproved extension of [Fav25, Thm 3.1] to all residue-characteristic-0 fields; without a repelling type I periodic point, the rigidity proof fails.","rationale":"The reader identified the same weakest assumption: Lemma 5.6 depends on an unproved extension of a cited theorem. I examined the surrounding arguments and found no additional load-bearing flaw that would change the verdict. The proof of Lemma 5.6 is otherwise logically coherent: the nested-disk argument uses the fixed point to get f(∩D_n)=∩D_n and the contradiction from Julia points being boundary points is valid; the length-transfer to Y_v uses Proposition 5.2(3) and Proposition 4.7(4) consistently. The later sections (Theorems 1.6, 1.7, 6.3, 6.4) rely on Theorem 1.1 but not on new unproved extensions of external results. The cited [Fav25, Theorem 3.1] is explicitly parenthetical, making the gap transparent and addressable. Since the gap is specific and plausibly fillable, a conditional acceptance is appropriate: the paper's main claim is not yet fully substantiated, but there is no evidence of a fundamental obstacle. Therefore I keep the reader's verdict unchanged.","tokens_in":53053,"tokens_out":12375,"duration_ms":115840,"concrete_test":"Check whether [Fav25, Theorem 3.1] as stated or its proof actually covers all algebraically closed complete non-Archimedean fields of residue characteristic 0, not just formal Laurent series. If not, prove directly that any degree-≥2 polynomial with bad reduction over such a field has a repelling type I periodic point, e.g., by adapting the classical argument in Benedetto (2019) or Rivera-Letelier. If such a proof cannot be supplied, Lemma 5.6 is unproved and the rigidity theorem does not currently hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rigidity theorem (Theorem 1.1) rests on Theorem 5.4, whose proof depends on Lemma 5.6. Lemma 5.6 asserts that for any polynomial f of degree at least 2 with bad reduction over an algebraically closed complete non-Archimedean field of residue characteristic 0, there exists a repelling type I periodic point p in the Julia set. The proof cites [Fav25, Theorem 3.1] but immediately acknowledges in a parenthetical that the original statement is only for polynomials over formal Laurent series, asserting without proof that the argument remains valid for the general fields considered here. This is a load-bearing gap: the existence of a fixed point p in the Julia set is used to construct nested sublevel sets D_n with diameters tending to 0, and then to transfer the small-diameter property to all disks D_w for w in Y_v via the tree length estimates. Without such a p, Lemma 5.6 collapses, and with it the coefficient comparison via Newton identities and Vieta's formulas in Theorem 5.4. This in turn invalidates Proposition 5.8 and the full equivalence in Theorem 1.1, and thereby the degeneration proof of Theorem 1.6/1.7 and the final uniformity bound (Theorem 1.2). The rest of the manuscript appears internally consistent, and the gap is plausibly fillable because the existence of repelling periodic points in the Julia set is a known phenomenon in non-Archimedean dynamics, but the present text does not provide the needed proof or a precise reference for this generality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a non-Archimedean analogue of Julia-set rigidity: over an algebraically closed complete non-Archimedean field of residue characteristic 0, if two polynomials have the same Julia set and one does not have potential good reduction, then they are dynamically related by an affine transformation and iteration (Theorem 1.1). This rigidity statement is then used, via an ultrafilter degeneration argument, to establish a uniform bound on the number of common preperiodic points of two complex polynomials in terms only of their degrees (Theorem 1.2), thereby proving the DeMarco–Krieger–Ye conjecture for polynomials. The paper also derives relative versions leading to special cases of the DeMarco–Mavraki conjecture (Theorems 1.3, 1.5, 7.1, 7.3). The proof introduces a non-Archimedean simplicial tree associated to a polynomial and uses it to compare preimage trees of f and g, combined with a hybrid-space degeneration technique.","tokens_in":53459,"tokens_out":5072,"duration_ms":54861,"significance":"If the main results hold, this is a substantial contribution. Theorem 1.1 is the first non-Archimedean Julia-set rigidity theorem for polynomials in this generality, and it gives a new proof avenue for the uniform common-preperiodic-point conjecture. The paper is also noteworthy for combining tree combinatorics, Berkovich potential theory, and ultrafilter degeneration in a way that yields explicit uniform constants. The non-circularity of the argument is a strength: the central results are not assumed, and the ultrafilter construction is an external tool. However, the paper is not yet in publishable form because one key lemma, on which the rigidity theorem depends, is asserted without proof in the required generality.","major_comments":[{"comment":"This is the load-bearing gap. The proof of Lemma 5.6 invokes [Fav25, Theorem 3.1] to obtain a repelling type I periodic point p in the Julia set of a polynomial without potential good reduction, but the original theorem is stated only for polynomials over formal Laurent series. The parenthetical '(Note: Although the original statement is formulated for polynomials over formal Laurent series, the proof remains valid for fields with residue characteristic 0.)' is not a proof. This existence is used to construct nested disks D_n of diameter tending to 0, which then transfer small-diameter preimages to all w∈Y_v. If this step fails, the coefficient comparison via Newton identities and Vieta's formulas in Theorem 5.4 collapses, and with it Proposition 5.8, the implication (2)⇒(1) in Theorem 1.1, and the degeneration arguments of Theorems 1.6, 1.7, and 1.2. The gap may be fillable, because rep","section":"§5.2, Lemma 5.6"},{"comment":"The proof of Theorem 1.7 is dismissed with 'The proof of Theorem 1.7 follows the same lines as that of Theorem 1.6, with only minor modifications.' This is not sufficient for a theorem that is used in Step 3 of Theorem 6.3 to control places of large residue characteristic. In particular, one must verify that the ultralimit field H(ω) has residue characteristic 0 when the residue characteristics of k_n are only assumed to be ≥ n, and that Lemma 6.1 (monic centered polynomial without potential good reduction) applies to H(ω). Since Theorem 1.7 supplies the local lower bound for all good places with char(\\tilde{k}_ν) ≥ C_5, a full proof or a detailed statement of which steps change is needed.","section":"§6.1, Theorem 1.7"},{"comment":"The proof of the claim that ∑_{ν good} N_ν ≥ 1/2 rests on a four-case pigeonhole argument with tightly chosen constants. The contradiction in case (3) uses h(f,g) ≥ h(a_i) ≥ C, but the constants 8CM and 8C are not tracked explicitly through the sums, and the role of the factor M = max_i deg p_i is not made precise. Because this claim is the bridge from local to global lower bounds and hence feeds directly into Theorem 1.2, it should be stated and proved as a separate lemma with all estimates displayed, rather than folded into a paragraph.","section":"§6.3, Theorem 6.4"}],"minor_comments":[{"comment":"The affiliation line reads 'Affliation: today.'; this is a clear typo that should be corrected.","section":"p.1"},{"comment":"The notation for |(f,g)| has mismatched braces: 'max{|a_i|,|b_j|,|b_{d2}|^{-1}|' should be 'max{|a_i|_ν, |b_j|_ν, |b_{d2}|_ν^{-1}}'. Also, in the proof of Theorem 1.6, equation (6.2) writes '= e' without explaining that e denotes the limit value of |(f_n,g_n)|^{ε_n}; clarify.","section":"§6.1, after Lemma 6.1"},{"comment":"The paragraph 'Before beginning the proof of Theorem 5.4' gives a heuristic example with f=z^2+c_1, g=z^2+c_2. It says 'Fix an ε > 0. We may then take n large enough so that f^{-n}(D(0,R))=g^{-n}(D(0,R)) is a disjoint union of 2^n disks, each of radius < ε.' This is illustrative but the notation ε is reused for the coefficient comparison; consider renaming to avoid confusion.","section":"§5.2, before Theorem 5.4"},{"comment":"Reference [Fav25] is cited as '[Fav25, Theorem 3.1]', but the bibliographic entry does not list theorem numbers. Since the lemma is load-bearing, please indicate the precise theorem or include the relevant statement.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an important conjecture and the overall architecture is convincing: tree combinatorics plus ultrafilters is a natural and promising route. The central problem is the unproved extension of [Fav25, Thm 3.1] in Lemma 5.6; this is not a cosmetic issue because the rigidity theorem and the uniformity consequence depend on it. The other sketched steps (Theorem 1.7, part of Theorem 6.4) are secondary but should be expanded. Given the plausibility of fixing Lemma 5.6, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. If the proof holds, it settles the DeMarco–Krieger–Ye conjecture for all complex polynomial pairs, allowing different degrees, and it proves a new non-Archimedean rigidity theorem for Julia sets. Both are real steps forward, not incremental.\n\nWhat's actually new: the uniform bound in Theorem 1.2 is not covered by prior work, which only handled unicritical families, Lattès maps, one-parameter families, or Zariski open subsets of moduli. The tree construction extends DeMarco–McMullen to the non-Archimedean setting, and the ultrafilter degeneration argument follows Favre–Gong. The paper is careful with the strategy, and the dependencies are spelled out.\n\nThe weak spot is Lemma 5.6. The proof invokes [Fav25, Theorem 3.1] as the source for a repelling type I periodic point in the Julia set, but that theorem is stated for formal Laurent series. The paper says in a parenthetical that the proof remains valid for residue-characteristic-0 fields, with no justification. This is load-bearing: without that point, the nested disks and the diameter estimates that feed into Theorem 5.4 don't work, and the rigidity theorem falls apart. The gap is plausibly fillable—repelling periodic points are known in non-Archimedean dynamics—but it needs a proof or a precise reference, not a parenthetical.\n\nMore minor: Theorem 1.7 is dismissed with 'the proof follows the same lines,' part of Theorem 6.4 is 'easily extends,' and the pigeonhole estimates in Theorem 6.3 are terse. These are stylistic; the main concern is Lemma 5.6.\n\nThe citation pattern looks fine. The self-citations to [FG25] and [Yap25] are for construction tools, not the target theorem. No circularity.\n\nWho should read it: arithmetic dynamicists working on unlikely intersections and non-Archimedean dynamics. It deserves a serious referee. I'd send it to review, but the referee request should ask for a complete proof or explicit reference for Lemma 5.6. Conditional accept is the right call.","headline":"A genuinely new result with a load-bearing gap: Lemma 5.6 needs a proof or a precise reference before the main theorems are fully grounded.","tokens_in":53896,"tokens_out":3366,"would_cite":true,"duration_ms":32812,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P05","37P30","37P45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any two complex polynomials of fixed degrees, either their preperiodic sets coincide or the overlap is uniformly bounded by a constant depending only on the degrees.","keywords":["non-Archimedean dynamics","Julia set rigidity","preperiodic points","uniform bounds","energy pairing","Berkovich space","polynomial dynamics","unlikely intersections"],"falsifier":"Find a counterexample: two polynomials over an algebraically closed complete non-Archimedean field of residue characteristic 0, neither with potential good reduction, with the same Julia set but no relation f=σ∘g^n; or two complex polynomials of fixed degrees with Prep(f)≠Prep(g) and arbitrarily many common preperiodic points. Alternatively, check the cited theorem's proof for validity in this generality, for instance by attempting to construct a repelling periodic point in a generic field of residue characteristic 0 where the original argument does not apply.","tokens_in":52933,"feed_emoji":"🔄","tokens_out":7003,"duration_ms":70558,"temperature":0.7,"pith_summary":"This paper establishes a rigidity theorem for polynomial dynamics over non-Archimedean fields of residue characteristic 0: if a polynomial whose Julia set is more than a single point shares that Julia set with any other polynomial, then the two maps are dynamically related — one is an iterate of the other composed with an affine symmetry. From this rigidity the paper derives a uniform bound for complex polynomials: given degrees d1,d2≥2, either two such polynomials have exactly the same set of preperiodic points, or that common set has size bounded by a constant depending only on the degrees. This settles the uniformity conjecture for common preperiodic points in the polynomial case, including pairs of different degrees. The paper also establishes relative versions, proving the relevant Zariski-density conjecture for fibered powers of the diagonal in polynomial moduli spaces.","feed_headline":"Polynomials share few preperiodic points unless they share all","feed_subtitle":"For any two complex polynomials, the overlap of preperiodic sets is either total or bounded by a number depending only on the degrees.","key_machinery":"The key object is the polynomial tree T_f, a simplicial tree inside the Berkovich projective line whose ends are the points of the Julia set; its vertices are the grand orbits of branch points of the convex hull of the Julia set and infinity, and the polynomial acts on it as a simplicial map. Equality of Julia sets forces T_f=T_g with the same vertex set. A minimal finite subset X_v with f^{-1}(X_v)=g^{-1}(X_v) then produces preimage disks of arbitrarily small diameter, using a repelling periodic point in the Julia set; comparing elementary symmetric functions of the preimages via Vieta's formulas and Newton's identities, with |n|=1 in residue characteristic 0, forces the coefficients to agr","core_discovery":"The central claim is Theorem 1.2: for any integers d1,d2≥2 there is a constant M(d1,d2) such that for any two complex polynomials f,g of degrees d1 and d2, either Prep(f)=Prep(g) or |Prep(f)∩Prep(g)|≤M. The proof rests on Theorem 1.1, a non-Archimedean rigidity statement: over an algebraically closed complete non-Archimedean field of residue characteristic 0, if a polynomial f without potential good reduction — equivalently, with Julia set not reduced to a single point — and a polynomial g have the same Julia set, then f=σ∘g^n for some n≥1 and affine σ preserving the Julia set. Same Julia set, same equilibrium measure, same Green function, and same Böttcher coordinate up to a root of unity a","pith_inferences":["The paper leaves the constant M(d1,d2) non-explicit; tracking the Vieta/Newton comparisons and the constants in the energy lower bounds could in principle yield an effective bound, which the authors do not state.","The same tree-based rigidity, if extended to rational maps along the lines the authors indicate, would likely imply a uniform bound for all rational maps of fixed degrees, not just polynomials.","The failure of rigidity in positive residue characteristic, illustrated by maps whose Julia set is the p-adic integers, suggests the uniform bound may also fail without the residue-characteristic-0 hypothesis; testing that family with the energy pairing would be informative.","The local energy-pairing lower bounds may have independent use in quantitative equidistribution of preperiodic points, since they give uniform separation of the equilibrium measures."],"forward_implications":["For complex polynomials, the common preperiodic set is either everything or at most a constant-size finite set, with the constant independent of the coefficients.","A polynomial over a residue-characteristic-0 non-Archimedean field is determined up to affine composition by its Julia set, equilibrium measure, Green function, or Böttcher coordinate.","The relative Bogomolov statement holds: for a family of polynomial pairs, points of small height in the fibered power of the diagonal are not Zariski dense unless the maps share all preperiodic points.","The analogous Zariski-density conjecture holds for fibered powers of the diagonal in the space of monic centered polynomial pairs."],"fun_headline_variants":["Shared preperiodic points: all or bounded","Preperiodic sets: equal or small overlap","Same Julia set implies dynamical link in p-adics","Common preperiodic points: total or degree-capped","Rigidity theorem: preperiodic overlap all-or-few"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction of arbitrarily small preimage disks uses a theorem on repelling periodic points imported from Laurent-series fields and assumes, without proof, that it holds over every algebraically closed complete non-Archimedean field of residue characteristic 0; if that extension fails, the rigidity theorem and the uniform bound collapse.","fun_headline_variants_meta":{"raw":{"variants":["Shared preperiodic points: all or bounded","Preperiodic sets: equal or small overlap","Same Julia set implies dynamical link in p-adics","Common preperiodic points: total or degree-capped","Rigidity theorem: preperiodic overlap all-or-few"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1412,"prompt_tokens":727,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":607}},"tokens_in":471,"tokens_out":685,"duration_ms":7382,"temperature":1.0,"reasoning_tokens":607,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:55:58.241222+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a counterexample: two polynomials over an algebraically closed complete non-Archimedean field of residue characteristic 0, neither with potential good reduction, with the same Julia set but no relation f=σ∘g^n; or two complex polynomials of fixed degrees with Prep(f)≠Prep(g) and arbitrarily many common preperiodic points. Alternatively, check the cited theorem's proof for validity in this generality, for instance by attempting to construct a repelling periodic point in a generic field of residue characteristic 0 where the original argument does not apply.","supporting_citations":[],"review_version":1}