On the Exceptional Sets of p-adic Transcendental Analytic Functions

2024/07/17 by Bruno de Paula Miranda, Miranda, Bruno De Paula, Jean Lelis +1
Mathematics · #11J61 #11S80 #FOS: Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.13015

openalex publication_date 2024/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ⊆ ℚ ∩ B(0, ρ) 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 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 ∈ ℚ_ρ[[z]] such that S_f = S. Furthermore, if ρ≥ 1, f can be taken in ℤ_ρ[[z]]. Additionally, we demonstrate that any S ⊆ ℚ ∩ B(0, ρ) containing 0 can be the exceptional set of uncountably many transcendental analytic functions f ∈ ℚ_ρ[[z]].

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