HAYMAN’S CONJECTURE IN A p-ADIC FIELD

Abstract

In this paper we study the famous Hayman's conjecture for transcendental meromorphic functions in a $p$-adic field by using methods of $p$-adic analysis and particularly the $p$-adic Nevanlinna theory. In $\mathbb{C}$, W. K. Hayman's stated that if $f$ is a transcendental meromorphic function, then $f' + af^m$ has infinitely many zeros that are not zeros of $f$ for each integer $m \geq 3$ and $a \in \mathbb{C} \setminus \{0\}$, which was proved in [2], [6], [8] and [11]. Here we examine the problem in an algebraically closed complete ultrametric field $\mathbb{K}$ of characteristic zero. Considering the function $f' + Tf^m$ with $T \in \mathbb{K}(x)$, we show that Hayman's statement holds for each $m \geq 5$ and $m = 1$. Further, if the residue characteristic of $\mathbb{K}$ is zero, then the statement holds for each positive integer $m$ different from $2$. We also examine the problem inside an ``open'' disc.