A characterization of non-Archimedean entire functions of bounded index (Q1355408)

In this paper some properties of complex entire functions are compared with entire functions over a field \(K\), which is algebraically closed, has characteristic zero and is complete w.r.t. a non-Archimedean valuation. An entire function \(f\) is called of index \(\leq k\) if for every \(z\in K\) and every \(j>k\) one has \(| \frac{f^{(j)}(z)}{j!}| \leq \max _{0\leq i\leq k}| \frac{f^{(i)}(z)}{i!}| \). It is shown that this property is equivalent to: \(f\) at most \(k\) zeros (counted with multiplicity) in every open disk with radius 1. The growth of an entire function of index \(\leq k\) (over \(K\)) is however not exponential as in the complex case. Generalized derivatives and corresponding generalized indices for entire functions over \(K\) are introduced and the result is extended.