Search

Following a previous paper by Jacqueline Ojeda and the first author, here we examine the number of possible branched values and branched functions for certain $p$-adic and complex meromorphic functions where numerator and denominator have different kind of growth, either when the denominator is small comparatively to the numerator, or vice-versa, or (for p-adic functions) when the order or the type of growth of the numerator is different from this of the denominator: this implies that one is a small function comparatively to the other. Finally, if a complex meromorphic function $\displaystyle{f\over g}$ admits four perfectly branched small functions, then $T(r,f)$ and $T(r,g)$ are close. If a p-adic meromorphic function $\displaystyle{f\over g}$ admits four branched values, then $f$ and $g$ have close growth. We also show that, given a p-adic meromorphic function $f$, there exists at most one small function $w$ such that $f-w$ admits finitely many zeros and an entire function admits no such a small function.

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables., Comment: 20 pages, 0 figures, LaTeX, to appear in Journal of Number Theory

Let K be a complete algebraically closed p-adic field of characteristic zero. Let f, g be two transcendental meromorphic functions in the whole field K or meromorphic functions in an open disk that are not quotients of bounded analytic functions. Let P be a polynomial of uniqueness for meromorphic functions in K or in an open disk and let $\alpha$ be a small meromorphic function with regards to f and g. If f'P'(f) and g'P'(g) share $\alpha$ counting multiplicity, then we show that f=g provided that the multiplicity order of zeroes of P' satisfy certain inequalities. If $\alpha$ is a Moebius function or a non-zero constant, we can obtain more general results on P., Comment: Ici on trouve les r\'esultats sans les d\'emonstrations d'un article de 30 pages

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.

The proof of the intermediate value theorem for power series on a Levi-Civita field will be presented. After reviewing convergence criteria for power series [19], we review their analytical properties [18]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iteration, we construct a sequence that converges strongly to a point at which the intermediate value will be assumed.