On the Exceptional Sets of $p$-adic Transcendental Analytic Functions

TL;DR

This paper investigates the structure of exceptional sets of $p$-adic transcendental analytic functions, establishing conditions for their possible forms and constructing functions with prescribed exceptional sets.

Contribution

It provides necessary conditions for exceptional sets and constructs uncountably many transcendental functions with given exceptional sets under certain symmetry and containment conditions.

Findings

Necessary conditions for exceptional sets of $p$-adic transcendental functions.

Existence of uncountably many functions with prescribed exceptional sets.

Construction of functions with arbitrary exceptional sets containing 0.

Abstract

In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S \subseteq \overline{\mathbb{Q}} \cap B(0, \rho)$ to be the exceptional set of a $p$-adic transcendental analytic function with rational coefficients, demonstrating that, in general, the answer to Mahler's Problem C over $\mathbb{C}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f \in \mathbb{Q}_{\rho}[[z]]$ such that $S_f = S$. Furthermore, if $\rho \geq 1$, $f$ can be taken in $\mathbb{Z}_{\rho}[[z]]$. Additionally, we demonstrate that any $S \subseteq \overline{\mathbb{Q}} \cap B(0, \rho)$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f \in \overline{\mathbb{Q}}_{\rho}[[z]]$.