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.