abstract 81
Bulletin of Computational Applied Mathematics (Bull CompAMa)
81
The p-adic Hayman conjecture. A survey with some generalizations
Alain Escassut
Let ${\rm I\!K}$ be a complete ultrametric algebraically closed field of characteristic $0$. According to the $p$-adic Hayman conjecture applied to transcendental meromorphic function $f$ in ${\rm I\!K}$ or an "unbounded" meromorphic function inside an open disk, for each $n\in {\rm I\!N}^*$,\ $f^nf'$ takes every value $b\neq 0$ infinitely many times. That was proved for $n\geq 3$ by J. Ojeda and next, by herself and the author for meromorphic function in ${\rm I\!K}$, for $n=2$. Here we recall these proofs and generalize them to meromorphic functions out of a hole whenever $n\geq 3$. We also recall the proof of this theorem: given a meromorphic function $f$, there exists at most one small function $w$ such that $f-w$ have finitely many zeros.
Keywords: p-adic meromorphic functions; value distribution; exceptional values.
Cite this paper:
Escassut A.
The p-adic Hayman conjecture. A survey with some generalizations.
Bull. Comput. Appl. Math. (Bull CompAMa)
Vol. 11, No.1 pp.213-248, 2023.