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A digit-sum formula for Böttcher coordinates

This paper establishes an unconditional digit-sum congruence formula for the coefficients of the Böttcher coordinate associated with the polynomial $\phi(x)=x^{p^2}+p^2x^{p^2+1}$ over an odd prime $p$ , thereby confirming conjectures by Salerno and Silverman and providing an explicit description of the first block of coefficients modulo $p$ .

Original authors: Rufei Ren

Published 2026-04-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rufei Ren

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery hidden inside a giant, infinite mathematical machine. This machine is called a Böttcher coordinate.

In the world of math, some machines (called functions) take a number, do something to it, and spit out a new number. If you keep feeding the output back into the machine, the numbers usually get huge and run away to infinity. The "Böttcher coordinate" is like a special map or a secret decoder ring that translates this chaotic, running-away behavior into something simple and predictable, like a straight line or a simple power.

The author of this paper, Rufe Ren, is looking at a very specific, tricky machine defined by the formula:
$\phi(x) = x^{p^2} + p^2 x^{p^2+1}$
Here, $p$ is a special kind of number called an odd prime (like 3, 5, 7, 11...). The machine is built using these primes.

The Mystery: The Hidden Pattern

When you try to decode this machine, you get a long list of numbers (coefficients) that make up the secret map. Let's call these numbers $a_1, a_2, a_3, \dots$ .

For a long time, mathematicians Salerno and Silverman noticed something weird about these numbers when you look at them through a specific lens (called "modulo $p$ "). They saw patterns, like a repeating rhythm, but they couldn't prove why the rhythm existed. They made a guess (a conjecture) about what the numbers would be, but they needed a proof.

The Detective's Toolkit: Digit Sums

Rufe Ren's breakthrough is realizing that the secret to these numbers isn't just the number itself, but how the number is written in base $p$ .

Think of our normal numbers as being written in Base 10 (using digits 0-9). The number 123 means $1 \times 100 + 2 \times 10 + 3 \times 1$ .
The "Digit Sum" is just adding those digits up: $1 + 2 + 3 = 6$ .

Ren discovered that for this specific machine, the value of the secret number $a_k$ depends entirely on the sum of the digits of $k$ when written in Base $p$ .

The Analogy: The "Carry" Game

To understand the proof, imagine a game of adding numbers where you are only allowed to use digits from 0 to $p-1$ .

The Problem: When you add two numbers, sometimes the sum is too big for a single digit, so you have to "carry" a 1 over to the next column (like $9+1=10$ , you write 0 and carry the 1).
The Insight: Ren realized that the "messy" parts of the calculation (the parts that usually make math hard) cancel each other out perfectly, unless there are these "carries."
The Result: The number of carries and the sum of the digits act like a filter. Most potential answers get filtered out (they become zero or vanish). Only the answers that fit a very specific "digit-sum" pattern survive.

The Three Main Discoveries

1. The "Zero" Case (The Easy Mode)
If the number $k$ is a multiple of $p$ (like $p, 2p, 3p$ ), the pattern is surprisingly simple. The secret number $a_k$ is just $+1$ or $-1$, alternating like a heartbeat.

Analogy: It's like a light switch. If you press it once, it's off. Twice, it's on. Three times, off. No complicated math needed.

2. The "Non-Zero" Case (The Hard Mode)
If $k$ isn't a multiple of $p$ , the pattern is a bit more complex but still follows a strict rule based on the digit sum.

The Formula: The paper gives a formula that says: "Take the digit sum, raise a few numbers to that power, and multiply them."
The Magic: Ren proved that you don't need to calculate the whole machine to find the answer. You just need to look at the digits of the position $k$ , add them up, and plug them into this simple formula.

3. Solving the Conjecture
The paper proves that the guesses made by Salerno and Silverman were 100% correct.

They guessed that numbers at positions $p^m$ follow a simple $(-1)^m$ pattern. Proven.
They guessed that numbers just before those positions are zero. Proven.
They guessed that numbers two spots before are always $-1$. Proven.

Why Does This Matter?

In the real world, this might seem like abstract number crunching. But in the world of Non-Archimedean Dynamics (a branch of math that studies how things behave in strange, "p-adic" number systems), this is huge.

Think of it like finding a universal law of physics for a specific type of universe. Before this paper, we knew the universe had a rhythm, but we didn't know the sheet music. Ren has written down the sheet music.

In Summary:
Rufe Ren took a chaotic, infinite mathematical machine, realized that its secret code is written in the sum of digits, and proved that the code follows a beautiful, predictable pattern. He turned a complex, unsolved mystery into a simple, elegant formula that anyone (with a calculator and a pencil) can check.

1. Problem Statement

The paper addresses the arithmetic properties of the coefficients of the Böttcher coordinate for a specific family of polynomials over $p$ -adic rings.

Context: In complex and non-archimedean dynamics, the Böttcher coordinate $f_\phi(x)$ linearizes a map $\phi(x)$ near a superattracting fixed point (usually at infinity) via the functional equation $\phi(f_\phi(x)) = f_\phi(x^q)$ , where $q$ is the degree of the leading term.
Specific Object: The author studies the polynomial $\phi(x) = x^{p^2} + p^2 x^{p^2+1}$ over the $p$ -adic integers, where $p \ge 3$ is an odd prime. Here, the degree is $q = p^2$ .
The Question: The Böttcher coordinate is a power series $f_\phi(x) = x \sum_{k \ge 0} \frac{a_k}{k!} x^k$ . Previous work by Salerno and Silverman [8] observed striking patterns in the coefficients $a_k$ modulo $p$ and formulated Conjecture 25, predicting specific residue-class behaviors for coefficients of the form $a_{p^m}$ , $a_{p^m-1}$ , and $a_{p^m-2}$ .
Goal: To prove an unconditional, explicit formula for $a_k \pmod p$ based on the base- $p$ digit sum of $k$ , thereby resolving the conjecture and providing a complete description of the coefficients.

2. Methodology

The proof employs a sophisticated combination of combinatorial analysis, $p$ -adic valuation theory, and generating function techniques.

A. The A-B-C Recursion

The author derives a recursive relation for the coefficients $a_k$ by comparing coefficients in the functional equation $\phi(f_\phi(x)) = f_\phi(x^q)$ . The relation is decomposed into three parts:
$a_k = A_k[x^k] - B_k[x^k] - C_k[x^k]$
where $A_k, B_k, C_k$ are derived from the expansion of $\phi(f_\phi(x))$ .

Proposition 2.1: Proves that the $A$ -term vanishes modulo $p$ for all $k \ge 1$ .
Proposition 2.2: Analyzes the $C$ -term. It shows that if $k \equiv a \pmod p$ ( $1 \le a \le p-1$ ), then $C_k[x^k] \equiv a a_{k-1} \pmod p$ . If $p \mid k$ , the $C$ -term vanishes.

B. Analysis of the B-term (Combinatorial Core)

The $B$ -term involves a multinomial expansion. The author analyzes the $p$ -adic valuation of the coefficients of monomials in this expansion using Legendre's Formula ( $\text{ord}_p(n!) = \frac{n - S_p(n)}{p-1}$ ).

Carry Defects: The valuation is expressed in terms of "carry defects" $c(e)$ and $d(e)$ arising from base- $p$ additions.
Surviving Monomials: A monomial survives modulo $p$ $p$ only if the total defect is minimal.
- Case $p \mid k$ : The author proves that surviving monomials must have indices divisible by $p$ , except for a specific "exceptional" monomial involving $a_0$ and $a_1$ . This leads to a recursive structure depending only on earlier coefficients $a_{p^i}$ .
- Case $p \nmid k$ : The author introduces Vector Partitions. The indices are encoded as vectors of base- $p$ digits. The condition for survival forces the addition of indices to be "carry-free" in base $p$ .

C. Generating Functions and Cumulants

To handle the complex combinatorics of the vector partitions for the general case ( $a \neq 0$ ):

Vector Partition Expansion: The $B$ -term is expanded as a sum over vector partitions of the digit vector of $k$ .
Inductive Reduction: Using an induction hypothesis on the residue class $a$ and the digit sum $s$ , the author shows that the block polynomials in the expansion depend only on the total weight of the digit vectors.
Multivariate to Univariate Collapse: The author defines a multivariate moment series and its associated cumulants. A key technical breakthrough (Proposition 5.4) is proving that the multivariate cumulant collapses into a univariate expression involving the operator $N = y \frac{d}{dy}$ acting on a truncated exponential series. This reduces a high-dimensional combinatorial sum to a single-variable calculation.

3. Key Contributions and Results

The Main Theorem (Digit-Sum Formula)

The paper establishes a closed-form congruence for the coefficients $a_k$ . Let $k = a + d_1 p + \dots + d_r p^r$ with $0 \le a, d_i \le p-1$ , and let $s = \sum d_i = S_p(k) - a$ .
Theorem 1.1:
$a_k \equiv (-1)^s (a+1)^s a^a \pmod p$
A stronger, unified form is also provided:
$a_k \equiv (-1)^{S_p(k)} (a+1)^{S_p(k)-1} \pmod p$
(Note: The paper notes that for $a=0$ , the formula simplifies to $a_k \equiv (-1)^{S_p(k)} \pmod p$ .)

Resolution of Conjectures

The paper proves Salerno and Silverman's Conjecture 25 (Theorem 1.2):

$a_{p^m} \equiv (-1)^m \pmod p$
$a_{p^m-1} \equiv 0 \pmod p$
$a_{p^m-2} \equiv -1 \pmod p$

Structural Insights

Divisible Case ( $a=0$ ): The coefficients $a_{p^m}$ are determined purely by a truncated exponential series identity, independent of the specific structure of $p^m$ beyond its divisibility.
Non-Divisible Case ( $a \neq 0$ ): The coefficients are governed by the interaction between the base- $p$ digit sum and the residue class $a$ , mediated by the vector partition structure.

4. Significance

Unconditional Proof: The results are proven unconditionally for all odd primes $p \ge 3$ , removing the need for the heuristic or conditional arguments often found in $p$ -adic dynamics.
Explicit Arithmetic Description: The paper provides the first explicit, computable description of the first block of coefficients modulo $p$ for this class of maps, moving beyond qualitative integrality results.
Methodological Innovation: The technique of reducing a multivariate partition problem to a univariate cumulant via vector partitions and carry-free addition is a novel contribution to the combinatorics of $p$ -adic power series.
Dynamics Implications: By characterizing the coefficients of the Böttcher coordinate, the paper sheds light on the integrality and convergence properties of these coordinates in non-archimedean dynamics, which are crucial for understanding the geometry of Julia sets and the Mandelbrot set in the $p$ -adic setting.

In summary, Rufe Ren's paper successfully bridges complex dynamics, combinatorics, and number theory to solve a specific, long-standing conjecture regarding the arithmetic nature of Böttcher coordinates, providing a powerful new formula based on the digit sums of the indices.