arxiv: 2604.02014 · v3 · submitted 2026-04-02 · 🧮 math.NT · math.DS

Recognition: 2 theorem links

· Lean Theorem

Coefficient-Level B\"ottcher Theory for Wild Superattracting Germs of Degree p^e

Rufei Ren

Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:35 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords Böttcher coordinatewild superattracting germsp-adic dynamicsdigit weightradius of convergencefunctional equationpure-power recursion

0 comments

The pith

The inverse Böttcher coordinate for the wild family φ_{r,e} has radius of convergence p^{-θ_{r,e}} determined by digit-weight bounds and a pure-power recursion.

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

The paper analyzes the inverse Böttcher coordinate f_{r,e}(x) for the family of maps φ_{r,e}(x) = x^{p^e} + p^{r+e} x^{p^e+1}. For the special fiber r=0 it establishes a complete mod-p digit-sum law on the coefficients. For r greater than or equal to 1 it proves a global lower bound on digit weights together with a leading monomial theorem, a lag-e recursion for pure powers, and subadditivity, which together fix the branch word and the radius. This radius formula gives the precise p-adic domain where the coordinate converges. The work also shows that sufficiently divisible tails preserve the leading behavior.

Core claim

In this family the inverse Böttcher coordinate is characterized by the functional equation φ(f(x)) = f(x^q) with q = p^e. Its coefficients satisfy a digit-weight lower bound that is global, a theorem identifying the leading monomial in divisible non-pure classes, a recursion of lag e on pure-power terms, and subadditivity of the induced weight. These facts determine the pure-power branch word (B^{e-1}A)^{⌈r/e⌉}B^∞ and produce the explicit radius ρ(f_{r,e}) = p^{-θ_{r,e}} where θ_{r,e} = p^{-e⌈r/e⌉} (1/(p-1) + e⌈r/e⌉ - r). Tails with valuation at least Λ_{r,e}(h+1) + 1 preserve the initial terms, the word, the asymptotic, and the radius.

What carries the argument

The inverse Böttcher coordinate f_{r,e}(x) = x ∑ a_k(r,e) x^k / k! satisfying the functional equation, together with the digit-weight valuation function on its coefficients that obeys the lower bound, leading-monomial, lag-e recursion, and subadditivity rules.

If this is right

The radius of convergence is exactly p to the power of minus θ_{r,e} for each fiber r and e.
For the clean fiber r=0 the coefficients obey a complete mod-p digit-sum law.
The pure-power branch word is (B^{e-1}A) raised to ⌈r/e⌉ followed by infinite B's.
Under the tail condition v_p(θ_h) ≥ Λ_{r,e}(h+1)+1 the leading terms and radius remain unchanged.
For e=2 the results recover the known radius statements for the degree-p^2 family.

Where Pith is reading between the lines

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

The digit-weight technique may extend to other superattracting germs not of this exact polynomial form.
Similar recursions could yield convergence radii for Böttcher coordinates in higher-dimensional p-adic dynamics.
Explicit computation of coefficients for small primes would test the predicted valuations directly.
The branch-word combinatorics might connect to formal group laws or other p-adic series expansions.

Load-bearing premise

The inverse Böttcher coordinate exists as a power series satisfying the functional equation, and its coefficients' p-adic valuations follow the digit-weight rules used to derive the bounds and recursion.

What would settle it

Compute the first several coefficients of f_{r,e} for p=3, e=2, r=1 by solving the functional equation recursively and check whether their valuations match the predicted θ_{1,2} and whether the series radius is exactly 3^{-θ}.

read the original abstract

Let $p$ be an odd prime, let $e\ge2$, and put $q=p^e$. We study the wild family \[ \varphi_{r,e}(x)=x^q+qp^r x^{q+1}=x^{p^e}+p^{r+e}x^{p^e+1} \qquad (r\ge0), \] and the inverse B\"ottcher coordinate $f_{r,e}(x)=x\sum_{k\ge0}a_k(r,e)x^k/k!$ characterized by \[ \varphi_{r,e}(f_{r,e}(x))=f_{r,e}(x^q). \] For the clean family, we prove a complete mod-$p$ digit-sum law in the special fiber $r=0$. For the higher fibers $r\ge1$, we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-$e$ pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word \[ (B^{e-1}A)^{\lceil r/e\rceil}B^\infty \] and the radius formula \[ \rho(f_{r,e})=p^{-\theta_{r,e}},\qquad \theta_{r,e}=p^{-e\lceil r/e\rceil}\left(\frac{1}{p-1}+e\lceil r/e\rceil-r\right). \] We then prove a tail-stable extension. In the special fiber, $p$-divisible tails preserve the digit-sum law modulo $p$. In the higher fibers, tails satisfying $v_p(\vartheta_h)\ge\Lambda_{r,e}(h+1)+1$ lie beyond the clean-family initial $\Lambda_{r,e}$-graded term and therefore preserve the leading terms, the pure-power branch word, the valuation asymptotic, and the radius. For $e=2$, this recovers the Salerno--Silverman degree-$p^2$ family and the Fu--Nie radius statement for the inverse coordinate in that family.

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.

Desk Editor's Note private letter to a colleague

The paper pins down explicit coefficient bounds and a radius formula for the inverse Böttcher coordinate in this wild p-adic family, extending the e=2 cases with a clean set of digit-weight tools.

read the letter

The main advance is the coefficient-level control for the inverse Böttcher coordinate f_{r,e} in the family φ_{r,e}(x) = x^{p^e} + p^{r+e} x^{p^e+1}. For the clean fiber r=0 they get a complete mod-p digit-sum law. For r≥1 they prove a global digit-weight lower bound, a leading-monomial theorem on divisible non-pure classes, a lag-e recursion on pure powers, and subadditivity of the induced weight. These combine to give the exact branch word (B^{e-1}A)^{⌈r/e⌉} B^∞ and the radius ρ(f_{r,e}) = p^{-θ_{r,e}} with the stated θ. The tail-stable extension then shows that sufficiently high-valuation perturbations preserve the leading terms and the radius. It recovers the Salerno-Silverman and Fu-Nie statements for e=2 without extra work, which is a good sign the setup is consistent with earlier results. The functional equation and p-adic valuation rules are used directly, with no free parameters or fitted data. The potential weak point is whether subadditivity plus the recursion and leading-monomial theorem really force equality only on the claimed arithmetic progressions and rule out other configurations that could produce the same or slower valuation growth. The abstract presents the four ingredients as sufficient, so the proofs presumably close the equality cases tightly; that step is the one a referee would check most carefully. The r=0 case looks more direct from the functional equation alone. This is for people working in arithmetic dynamics who need concrete radius or coefficient asymptotics for iteration. A reader who wants explicit formulas or who already knows the e=2 literature will get immediate value. It is worth sending to peer review because the claims are specific and the ingredients are stated clearly enough for verification.

Referee Report

1 major / 2 minor

Summary. The manuscript develops coefficient-level Böttcher theory for the wild family of superattracting germs φ_{r,e}(x) = x^{p^e} + p^{r+e} x^{p^e+1} (p odd prime, e≥2, q=p^e, r≥0). It proves a complete mod-p digit-sum law for the coefficients of the inverse Böttcher coordinate f_{r,e} in the special fiber r=0. For r≥1 it establishes a global digit-weight lower bound, a leading-monomial theorem on divisible non-pure classes, a lag-e pure-power recursion, and subadditivity of the induced digit weight; these are combined to identify the pure-power branch word (B^{e-1}A)^{⌈r/e⌉}B^∞ and to derive the explicit radius ρ(f_{r,e})=p^{-θ_{r,e}} with θ_{r,e}=p^{-e⌈r/e⌉}(1/(p-1)+e⌈r/e⌉-r). A tail-stable extension is proved that preserves the leading terms and radius under suitable valuation conditions on tails, recovering the e=2 results of Salerno-Silverman and Fu-Nie.

Significance. If the coefficient-level results hold, the paper supplies the first explicit, parameter-free radius formulas and valuation asymptotics for inverse Böttcher coordinates in this wild family of arbitrary degree p^e. The derivation rests directly on the functional equation together with p-adic digit-weight properties, without fitted parameters, and yields falsifiable predictions for the branch word and θ_{r,e}. This advances p-adic dynamics by giving precise control over the domain of the coordinate and provides a template for similar functional equations.

major comments (1)

[coefficient-level theorem for r≥1] In the coefficient-level theorem for r≥1 (the four ingredients: global digit-weight lower bound, leading monomial theorem, lag-e recursion, and subadditivity), the passage from these properties to the exact branch word (B^{e-1}A)^{⌈r/e⌉}B^∞ is not fully explicit. Subadditivity supplies only w(n+m)≤w(n)+w(m); to force equality on the arithmetic progressions required for the infinite B^∞ tail and the precise finite prefix of A's, the argument must verify that the leading-monomial theorem and lower bound exclude all other configurations. A concrete inductive check or equality-case analysis for the relevant residue classes would remove the risk that other digit sequences satisfy the stated conditions while producing a different asymptotic valuation growth.

minor comments (2)

[abstract] The abstract introduces the terms 'clean family' and 'higher fibers' without a brief definition or cross-reference; adding a parenthetical clarification would improve immediate readability.
[introduction] The branch-word notation (B^{e-1}A) and the symbols A, B are used in the abstract before they are defined; ensure the introduction supplies the definition of these letters (presumably corresponding to digit classes) at first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestion to strengthen the explicitness of the coefficient-level theorem. We address the major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: In the coefficient-level theorem for r≥1 (the four ingredients: global digit-weight lower bound, leading monomial theorem, lag-e recursion, and subadditivity), the passage from these properties to the exact branch word (B^{e-1}A)^{⌈r/e⌉}B^∞ is not fully explicit. Subadditivity supplies only w(n+m)≤w(n)+w(m); to force equality on the arithmetic progressions required for the infinite B^∞ tail and the precise finite prefix of A's, the argument must verify that the leading-monomial theorem and lower bound exclude all other configurations. A concrete inductive check or equality-case analysis for the relevant residue classes would remove the risk that other digit sequences satisfy the stated conditions while producing a different asymptotic valuation growth.

Authors: We agree that an explicit inductive verification would improve clarity and remove any ambiguity about uniqueness. In the revised manuscript we will add a dedicated subsection providing an equality-case analysis. The argument proceeds by induction on the number of p-adic digits: the global lower bound on digit weight together with the leading-monomial theorem on divisible non-pure classes immediately rules out any deviation from the prescribed finite prefix (B^{e-1}A)^{⌈r/e⌉}; once the prefix is fixed, the lag-e pure-power recursion forces every subsequent block to be a pure B-block, and subadditivity becomes equality on the resulting arithmetic progressions because any insertion of an A would violate the lower bound at the next valuation level. This establishes that no other digit sequence can satisfy the coefficient conditions of the functional equation, thereby confirming the branch word and the radius formula. revision: yes

Circularity Check

0 steps flagged

Derivation from functional equation and p-adic bounds is self-contained

full rationale

The paper establishes the coefficient-level theorem for r≥1 directly from the functional equation φ_{r,e}(f(x))=f(x^q) together with p-adic valuation rules. The four ingredients (global digit-weight lower bound, leading-monomial theorem on divisible non-pure classes, lag-e pure-power recursion, and subadditivity) are proved from these inputs and then combined to obtain the explicit branch word and θ_{r,e}. No parameter is fitted to data and then renamed a prediction; no self-citation supplies a uniqueness theorem that forces the result; the e=2 recovery of prior statements is presented as consistency check rather than load-bearing premise. The derivation therefore remains independent of its target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard p-adic analysis and the definition of the family; no new free parameters or invented entities apparent from abstract.

axioms (2)

standard math p-adic valuation properties and digit weight definitions in the context of power series coefficients
Used throughout the coefficient analysis for the Böttcher coordinate.
domain assumption Existence of the inverse Böttcher coordinate satisfying the functional equation
Assumed for the family φ_{r,e}.

pith-pipeline@v0.9.0 · 5692 in / 1624 out tokens · 57352 ms · 2026-05-15T07:35:19.821564+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For the higher fibers r≥1, we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-e pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word (B^{e-1}A)^{⌈r/e⌉}B^∞ and the radius formula ρ(f_{r,e})=p^{-θ_{r,e}}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel and Jcost_pos_of_ne_one unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ak ≡ (−1)^s (a+1)^s a^a (mod p) ... ord_p(ak(r)) = 1−p^{-r}/(p−1) k + O(1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

D. S. Alexander, F. Iavernaro, and A. Rosa, Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable during 1906–1942 , History of Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2011

work page 1906
[2]

Douady and J

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Première partie , Publica- tions Mathématiques d’Orsay 84–02, Université de Paris-Sud, Orsay, 1984

work page 1984
[3]

Douady and J

A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Deuxième partie , Publica- tions Mathématiques d’Orsay 85–04, Université de Paris-Sud, Orsay, 1985

work page 1985
[4]

Fu and H

H. Fu and H. Nie, Böttcher coordinates at wild superattracting ﬁxed points, Bull. Lond. Math. Soc. 56 (2024), no. 5, 1698–1715

work page 2024
[5]

J. H. Hubbard and P. Papadopol, Superattractive ﬁxed points in Cn, Indiana Univ. Math. J. 43 (1994), no. 1, 321–365

work page 1994
[6]

Milnor, Dynamics in One Complex Variable , 3rd ed., Annals of Mathematics Studies, vol

J. Milnor, Dynamics in One Complex Variable , 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006

work page 2006
[7]

J. F. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 21 (1920), 348–356

work page 1920
[8]

Salerno and J

A. Salerno and J. H. Silverman, Integrality properties of Böttcher coordinates for one-dimensional superattracting germs, Ergodic Theory Dynam. Systems 40 (2020), no. 1, 248–271

work page 2020
[9]

Gessel, Lagrange Inversion, J

I. Gessel, Lagrange Inversion, J. Combin. Theory Ser. A 144 (2016), 212–249

work page 2016
[10]

Surya and L

E. Surya and L. Warnke, Lagrange Inversion Formula by Induction, Amer. Math. Monthly 130 (2023), 944–948. Department of Mathematics, Fudan University Email address : rufeir@fudan.edu.cn

work page 2023