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 \(f \in \overline{\mathbb{Q}}_{\rho}[[z]]\) .
