Jacqueline Ojeda: Uniqueness for ultrametric analytic functions, p.153-165

Abstract:

Let Κ be a complete algebraically closed p-adic field of characteristic zero and let $f,\ g$ be two meromorphic functions inside an open disc of Κ. We first study polynomials of uniqueness for such functions. Suppose now $f,\ g$ are entire functions on Κ. Let a ∈ Κ \ {0} and $n, k \in \NN$N, with $k\geq 2$ and let $\alpha$ be a small entire function with respect to $f$ and $g$. If $f^n(f-a)^kf'$ and $g^n(g-a)^kg'$ share $\alpha$, counting multiplicities, with $n\geq \max\{ 6-k, k+1\}$ then $f=g$. If α ∈ Κ^* and if $n\geq \max\{ 5-k, k+1\}$ then $f=g$.

Let $f,\ g$ be unbounded analytic functions inside an open disk of  Κ and let $\alpha$ be a small function analytic inside in the same disk. If $f^n(f-a)^2f'$ and $g^n(g-a)^2g'$ share $\alpha$ counting multiplicities, with $n\geq 4$, then $f=g$. If $f^n(f-a)f'$ and $g^n(g-a)g'$ share $\alpha$ counting multiplicities, with $n\geq 5$, then $f=g$.

Key Words: Meromorphic, Nevanlinna, Ultrametric, Sharing Value, Unicity, Distribution of values..

2000 Mathematics Subject Classification: Primary: 12J25;
Secondary: 30D35; 30G06.

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