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A digit-sum formula for Böttcher coordinates

Rufei Ren

Abstract

Let $p$ be an odd prime and let $φ(x)=x^{p^2}+p^2x^{p^2+1}$. We prove an unconditional digit-sum congruence formula for the coefficients of the Böttcher coordinate of $φ$. The argument separates the divisible case $k\equiv 0\pmod p$ from the general case, gives a complete vector-partition description of the surviving $B$-term monomials, and reduces the remaining partition sum to a one-variable cumulant. As consequences we recover the residue-class congruences conjectured by Salerno and Silverman and obtain an explicit description of the first block of coefficients modulo $p$.

A digit-sum formula for Böttcher coordinates

Abstract

Let be an odd prime and let . We prove an unconditional digit-sum congruence formula for the coefficients of the Böttcher coordinate of . The argument separates the divisible case from the general case, gives a complete vector-partition description of the surviving -term monomials, and reduces the remaining partition sum to a one-variable cumulant. As consequences we recover the residue-class congruences conjectured by Salerno and Silverman and obtain an explicit description of the first block of coefficients modulo .

Paper Structure

This paper contains 7 sections, 16 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. The recursion and two structural lemmas
  3. The Case $a=0$
  4. Vector partitions and the $B$-term for $a\ne 0$
  5. Low coefficients and a multivariate cumulant collapse for $a\ne 0$
  6. Proof of \ref{['eq1']}
  7. Proof of \ref{['eq2']} and Theorem \ref{['thm:conj25']}

Key Result

Theorem 1.1

Let $k\ge 1$ and write with Then In fact, we have a stronger result $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (34)

  • Theorem 1.1: Digit-sum formula
  • Theorem 1.2: Salerno--Silverman's Conjecture 25
  • Proposition 2.1: The $A$-term always vanishes modulo $p$
  • proof
  • Proposition 2.2: A global $C$-term lemma
  • proof
  • Theorem 2.3: The case $a=0$
  • Lemma 3.1: A truncated-exponential coefficient identity
  • proof
  • Corollary 3.2
  • ...and 24 more