Digit-sum Formulas and Divisible Classes for Böttcher Coordinates of $x^{p^2}+p^{r+2}x^{p^2+1}

Abstract: Let $p$ be an odd prime and, for $r\ge 0$, let [ φ_r(x)=x^{p^2}+p^{r+2}x^{p^2+1}. ] For $r=0$ we prove a complete digit-sum congruence formula modulo $p$ for the coefficients of the Böttcher coordinate of $φ_0$. As consequences we prove Conjecture~25 of Salerno and Silverman and obtain an explicit description of the first block of coefficients modulo $p$. For $r\ge 1$ we prove on the divisible non-pure class a leading term theorem together with a global digit-weight lower bound. These results yield an unconditional proof of Conjecture~27(a) of Salerno and Silverman.

• The paper presents explicit digit-sum congruence formulas for coefficients of Böttcher coordinates in the p-adic setting, resolving Conjecture 25.
• It establishes sharp p-adic valuation lower bounds and leading term theorems for divisible classes when r ≥ 1.
• The study combines combinatorial digit expansion methods with p-adic analysis to advance understanding of non-archimedean dynamics.

Digit-Sum Congruences and Divisible Classes for Böttcher Coordinates of $x^{p^2}+p^{r+2}x^{p^2+1}$

Introduction

This paper provides a comprehensive arithmetic analysis of Böttcher coordinates over $\mathbb{Z}_p$ associated to the polynomial family $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$, where $p \geq 3$ is prime and $r \geq 0$. The main contributions are the complete explicit description of the mod $p$ structure of Böttcher coordinate coefficients for $r=0$, digit-sum congruence laws covering all residue classes, and sharp $p$-adic valuation results for the families with $r \geq 1$. The paper resolves Conjectures 25 and 27(a) from Salerno and Silverman concerning superattracting germs in the wild reduction case.

Böttcher Coordinates and Arithmetic Dynamics

Let $f_r(x) = x \sum_{k \geq 0} \frac{a_k(r)}{k!} x^k$ denote the Böttcher coordinate of $\mathbb{Z}_p$0, defined via the functional equation $\mathbb{Z}_p$1. In the non-archimedean context, the arithmetic structure of the coefficients $\mathbb{Z}_p$2 is central, with integrality, congruence, and valuation properties encoding subtle arithmetic dynamics information.

Mod $\mathbb{Z}_p$3 Digit-Sum Formula for $\mathbb{Z}_p$4

For $\mathbb{Z}_p$5, the paper establishes a full digit-sum congruence formula for every $\mathbb{Z}_p$6 modulo $\mathbb{Z}_p$7:

Given $\mathbb{Z}_p$8, $\mathbb{Z}_p$9, $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$0, the formula

$\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$1

holds. This congruence captures both the residue class and the digit sum structure of $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$2 within every modulo $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$3 class. The proof involves a delicate combinatorial analysis of the Böttcher recursion, projecting monomial contributions via digit expansions and leveraging base-$\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$4 carries and factorization properties.

Strong explicit results include:

• For $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$5, $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$6 (for $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$7), and similarly all coefficients on the divisible class.
• For all $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$8, $\phi_r(x) = x^{p^2} + p^{r+2} x^{p^2+1}$9, delivering a closed-form for the first block.
• The congruence $p \geq 3$0, $p \geq 3$1, $p \geq 3$2 (mod $p \geq 3$3), thus resolving Conjecture 25 of Salerno and Silverman.

These formulas are driven by an explicit recursive analysis at the coefficient level—$p \geq 3$4- and $p \geq 3$5-terms in the recursion vanish or are controllable via digit-sum combinatorics, and surviving $p \geq 3$6-monomials are characterized as vector partitions, with dependencies only on divisible lower-order coefficients.

Valuation Theory for $p \geq 3$7 and Divisible Classes

In the fibers with $p \geq 3$8, the core results concern precise lower bounds on the $p \geq 3$9-adic valuation of $r \geq 0$0, identification of main terms, and explicit branch patterns across the pure-power (i.e., $r \geq 0$1) and divisible non-pure classes ($r \geq 0$2, $r \geq 0$3 not a power of $r \geq 0$4):

• Define the digit-weight function $r \geq 0$5 for the base-$r \geq 0$6 digits of $r \geq 0$7.
• For $r \geq 0$8, $r \geq 0$9 not a pure power, the main result asserts the leading term theorem:

$p$0

where $p$1 is the $p$2-ary digit sum of $p$3.

• For all $p$4, the lower bound

$p$5

is proven, and for large indices the explicit asymptotic

$p$6

holds, confirming the expected linear growth and settling Conjecture~27(a) of Salerno and Silverman.

The techniques rely on a careful inductive organization of the Böttcher recursion, with simultaneous control of pure and divisible classes, strong subadditivity of the digit-weight, explicit handling of the pure-power branch structure, and digit-monomial projections. Pure-power coefficients exhibit alternating $p$7- or $p$8-branch dominance, and eventually stabilize in slope, matching the aforementioned linear law.

Methods: Combinatorics and Carry Structure

The proofs engage a deep combinatorial framework, including:

• Base-$p$9 expansion/carry defects and digit-sum tracking
• Recurrence decompositions into $r=0$0, $r=0$1, $r=0$2 terms with precise $r=0$3-adic estimation
• Vector partition encodings for indexing $r=0$4-term monomials
• Use of Lucas’ theorem, Legendre’s formula for $r=0$5-adic orders in multinomials, and explicit exponential series coefficient identities
• Cumulant expansions and reduction of multivariate recurrence to univariate cumulant evaluation

The structure of the recursion ensures that modulo $r=0$6, coefficients on a given residue class depend only on lower-order coefficients within specific divisible classes, allowing induction closures at all levels.

Implications and Future Directions

The unconditional solution of Conjectures 25 and 27(a) gives a template for analyzing arithmetic properties of Böttcher coordinates at wild superattracting fixed points for more general polynomial families beyond the model case studied. The precise description of $r=0$7-adic growth rates in terms of digit sums demonstrates new depth of arithmetic regularity not visible through naive estimates.

Practically, these results suggest strong constraints on the possible $r=0$8-adic behavior of Taylor expansions in non-archimedean dynamical systems, with potential applications to arboreal Galois representations, $r=0$9-adic Mandelbrot sets, and arithmetic equidistribution problems for preimages and periodic points. The combinatorial framework may be extensible to higher-dimensional non-archimedean systems or other “wild” Lie-type degenerations.

Theoretically, the methods invite similar analysis of digit-sum phenomena in other non-linear recursions associated to canonical coordinates in arithmetic geometry, as well as possible extension to positive characteristic function field analogs.

Conclusion

This paper completes the analysis of the arithmetic structure of Böttcher coordinates for the family $p$0. Via explicit digit-sum congruences, sharp valuation theorems, and resolution of open conjectures, the work substantially advances the interplay between digit combinatorics, $p$1-adic analysis, and non-archimedean dynamics. The methods set a foundational base for further arithmetic study of dynamical expansions in wild characteristic and related arithmetic dynamical systems.

Reference: "Digit-sum Formulas and Divisible Classes for Böttcher Coordinates of $p$2" (2604.02014)