Bruno De Paula Miranda

arXiv2025

On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients

In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $Δ(0;1)$, closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients.

math.NT↗

arXiv2024

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

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 $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ 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 $\mathbb{C}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f \in \mathbb{Q}_ρ[[z]]$ such that $S_f = S$. Furthermore, if $ρ\geq 1$, $f$ can be taken in $\mathbb{Z}_ρ[[z]]$. Additionally, we demonstrate that any $S \subseteq \overline{\mathbb{Q}} \cap B(0, ρ)$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f \in \overline{\mathbb{Q}}_ρ[[z]]$.

math.NT↗