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We investigate whether certain Diophantine equations have or have not solutions in entire or meromorphic functions defined on a non-Archimedean algebraically closed field of characteristic zero. We prove that there are no non-constant meromorphic functions solving the Erdös–Selfridge equation except when the corresponding curve is a conic. We also show that there are infinitely many non-constant entire solutions to the Markoff–Hurwitz equation. [ABSTRACT FROM PUBLISHER]

Letbe a complete ultrametric algebraically closed field of characteristic 0. According to thep-adic Hayman conjecture, given a transcendental meromorphic functionfin, for each,takes every valueinfinitely many times. It was proven by the second author for. Here, we prove it forby using properties of meromorphic functions having finitely many multiple poles. [ABSTRACT FROM PUBLISHER]