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Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.19252 v2 pith:3KLALDXC submitted 2026-07-21 math.DS math.NT

Chen Gong , Jit Wu Yap This is my paper · ORCID
classification math.DSmath.NT MSC 37P0537P3037P45
keywords non-ArchimedeandynamicsJuliasetrigiditypreperiodicpointsuniformboundsenergypairingBerkovichspacepolynomialunlikelyintersections
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

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§5.2, Lemma 5.6] 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
  2. [§6.1, Theorem 1.7] 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.
  3. [§6.3, Theorem 6.4] 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.
minor comments (4)
  1. [p.1] The affiliation line reads 'Affliation: today.'; this is a clear typo that should be corrected.
  2. [§6.1, after Lemma 6.1] 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.
  3. [§5.2, before Theorem 5.4] 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.
  4. [References] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found; central derivation is self-contained. The only flagged caveat is an unproved parenthetical generalization in Lemma 5.6, which is a correctness gap rather than a circularity.

full rationale

The derivation chain is not circular. Theorem 1.1's core implication J_f=J_g => f=σ∘g^n is proved by comparing polynomial trees (Propositions 5.1–5.2) and then performing coefficient comparison through preimage counting, Vieta's formulas and Newton's identities (Theorem 5.4). This proof never assumes the conclusion. Theorem 1.2 follows from Theorem 1.1 by a standard ultrafilter degeneration contradiction (Theorems 1.6, 1.7, 6.3, 6.4); the energy-pairing lower bounds are not fitted to the target bound. The self-citations [FG25] and [Yap25] supply only auxiliary ultraproduct facts (algebraic closedness, spherical completeness, continuity of pairings), not the rigidity statement, and they are parameter-free technical lemmas rather than a chain ending in the theorem being proved. The one passage that deserves flagging is Lemma 5.6: "By [Fav25, Theorem 3.1], f admits a repelling type I periodic point p" with the parenthetical "Although the original statement is formulated for polynomials over formal Laurent series, the proof remains valid for fields with residue characteristic 0." This is an unproved generalization and is load-bearing for the small-disk diameter argument; if it fails, Theorem 5.4 collapses. However, it is an external-result gap, not a case of the paper's output being identical to its input. There is no self-definitional step, no fitted parameter renamed as prediction, and no uniqueness claim imported from the authors' prior work to force the answer; the complex rigidity [SS95] and the commuting-polynomial classification [Rit20] are independent external inputs. Therefore the circularity score is minimal.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a substantial body of prior results. The only genuinely new ad hoc input is the unproved extension of [Fav25] in Lemma 5.6. The polynomial tree T_f is a constructed tool, not an unexplained postulated entity.

assumptions (6)
  • ad hoc to paper Existence of a repelling type I periodic point in the Julia set of a polynomial without potential good reduction over any algebraically closed complete non-Archimedean field of residue characteristic 0
    Invoked in Lemma 5.6; the cited [Fav25, Theorem 3.1] is stated for formal Laurent series, and the paper asserts without proof that it carries over.
  • standard math Schmidt–Steinmetz rigidity for complex polynomials: equality of Julia sets implies dynamical relation
    Used in Theorem 5.9 and in the archimedean part of Theorem 6.4.
  • standard math Favre–Rivera-Letelier theory: existence, properties, and energy pairing of equilibrium measures on the Berkovich projective line
    Foundational for Sections 2, 3, and 6.
  • standard math Ultrafilter product Banach ring construction: H(ω) is algebraically closed, spherically complete, with residue characteristic 0
    Used in proofs of Theorems 1.6 and 1.7; it is prior work by the authors/collaborators but a technical tool, not the target result.
  • standard math Yuan–Zhang theory of adelic line bundles and the potential bigness criterion [Yua24, Theorem 4.1]
    Used in Section 7 to derive relative results and the DeMarco–Mavraki special case.
  • standard math Gauthier's inequality [Gau24, Theorem B] relating the energy pairing to the number of common preperiodic points
    Used in the final step of Theorem 1.2.

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Pith. "Pith review of Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points." pith.science (2026). https://pith.science/paper/3KLALDXC

@misc{pith2026260719252,
  author       = {Pith},
  title        = {Pith review of: Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KLALDXC}},
  note         = {Machine review of arXiv:2607.19252}
}
abstract

Let $k$ be an algebraically closed, complete non-Archimedean field of residue characteristic $0$. Let $f$ be a polynomial of degree at least $2$ over $k$ which does not have potential good reduction. We prove that if $g$ is any other polynomial with the same Julia set, then $f$ and $g$ must be dynamically related. As a consequence, we show that for any two complex polynomials $f,g$ of degree at least $2$, either their sets of preperiodic points coincide, or the number of their common preperiodic points is uniformly bounded above by a constant depending only on the degrees, thereby answering a conjecture of DeMarco--Krieger--Ye for polynomials. We also establish relative results, allowing us to prove special cases of the DeMarco--Mavraki conjecture.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.