On p-adic meromorphic functions (Q796693)

Let D denote the open unit disc in \({\mathbb{C}}_ p ({\mathbb{C}}_ p\) denotes the completion of the algebraic closure of \({\mathbb{Q}}_ p)\). For a meromorphic function \(\phi\) on D and \(a\in {\mathbb{C}}_ p\) a certain real function \(T(\phi,a,t)\) of \(t>0\) is introduced which measures the set of zero's of \(\phi\)-a and a growth of the function \(\phi\). This function \(T(\phi,a,t)\) is an analogue of a function in Nevanlinna theory. It is shown that, under the condition that \(T(\phi,a,t)\) is unbounded, the function \(\phi\) is determined by the divisors \(\phi^{-1}(a_ 1)\), \(\phi^{-1}(a_ 2)\), \(\phi^{-1}(a_ 3)\) where \(a_ 1,a_ 2,a_ 3\) are three different points of \({\mathbb{C}}_ p\). We note that \(T(\phi,a,t)\) has no clear meaning in the p-adic case and that the theorem above can easily be proved under the weaker assumption that \(\phi\) is not the quotient of two bounded holomorphic functions on D.