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International audience; A new Nevanlinna theorem on q p-adic small functions is given. Let f, g, be two meromorphic functions on a complete ultrametric algebraically closed field IK of characteristic 0, or two meromorphic functions in an open disk of IK, that are not quotients of bounded analytic functions by polynomials. If f and g share 7 small meromorphic functions I.M., then f = g. Better results hold when f and g satisfy some property of growth. Particularly , if f and g have finitely many poles or finitely many zeros and share 3 small meromorphic functions I.M., then f = g. 1. Main results Let IK be a complete ultrametric algebraically closed field of characteristic 0. Let us fix a ∈ IK and let R ∈]0, +∞[. We denote by d(a, R −) the disk {x ∈ IK | |x − a| < R}. We denote by A(IK) the IK-algebra of entire functions in IK and by M(IK) the field of meromorphic functions which is its field of fractions. We denote by A(d(a, R −)) the IK-algebra of analytic functions in d(a, R −) i.e. the set of power series converging in the disk d(a, R −) and by M(d(a, R −)) the field of meromorphic functions in d(a, R −) i.e. the field of fractions of A(d(a, R −)). Moreover , we denote by A b (d(a, R −)) the IK-algebra of functions f ∈ A(d(a, R −)) that are bounded in d(a, R −), by M b (d(a, R −)) its field of fractions and we put M u (d(a, R −)) = M(d(a, R −)) \ A b (d(a, R −)). We define N (r, f) ([1], chapter 40 or [3], chapter 2) in the same way as for complex meromorphic functions [2]. Let f be a meromorphic function in all IK (resp. in d(0, R −)) having no zero and no pole at 0. Let (a n) n∈IN be the sequence of poles of f , of respective order s n , with |a n | ≤ |a n+1 | and, given r > 0, (resp. r ∈]0, R[), let q(r) be such that |a q(r) | ≤ r, |a q(r)+1 | > r. We then denote by N (r, f) the counting function of the zeros of f , counting multiplicity, as usual: for all r > 0, we put N (r, f) = q(r) j=0 s j (log |a j | − log(r)). Moreover, we denote by N (r, f) the counting function of the poles of f , ignoring multiplicity 0

International audience; Let E be a complete ultrametric space, let K be a perfect complete ultra-metric field and let A be a Banach K-algebra which is either a full K-subalgebra of the algebra of continuous functions from E to K owning all characteristic functions of clopens of E, or a full K-subalgebra of the algebra of uniformly continuous functions from E to K owning all characteristic functions of uniformly open subsets of E. We prove that all maximal ideals of finite codimension of A are of codimension 1. Introduction: Let E be a complete metric space provided with an ultrametric distance δ, let K be a perfect complete ultrametric field and let S be a full K-subalgebra of the K-algebra of continuous (resp. uniformly continuous) functions complete with respect to an ultrametric norm. that makes it a Banach K-algebra [3]. In [2], [4], [5], [6] we studied several examples of Banach K-algebras of functions and showed that for each example, each maximal ideal is defined by ultrafilters [1], [7], [8] and that each maximal ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here. Here we must assume that the ground field K is perfect, which makes that hypothesis necessary in all theorems.

Let IK be a complete ultrametric algebraically closed field and let A(IK) be the IK-algebra of entire functions on IK. For an f ∈ A(IK), similarly to complex analysis, one can define the order of growth as $$\rho \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {\log |f|\left( r \right)} \right)}}{{\log r}}$$ . When ρ(f) ≠ 0,+∞, one can define the type of growth as $$\sigma \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {|f|\left( r \right)} \right)}}{{{r^\rho }\left( f \right)}}$$ . But here, we can also define the cotype of growth as $$\psi \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{q\left( {f,r} \right)}}{{{r^\rho }\left( f \right)}}$$ where q(f, r) is the number of zeros of f in the disk of center 0 and radius r. Many properties described here were first given in the Houston Journal, but new inequalities linking the order, type and cotype are given in this paper: we show that ρ(f)σ(f) ≤ ψ(f) ≤ eρ(f)σ(f). Moreover, if ψ or σ are veritable limits, then ρ(f)σ(f) = ψ(f) and this relation is conjectured in the general case. Several other properties are examined concerning ρ, σ, ψ for f and f’. Particularly,we show that if an entire function f has finite order, then $$\frac{{f'}}{{{f^2}}}$$ takes every value infinitely many times.

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.

International audience; Let IK be an algebraically closed p-adic complete field of characteristic zero. We define the order of growth ρ(f) and the type of growth σ(f) of an entire function f (x) = ∞ n=0 anx n on IK as done on l C and show that ρ(f) and σ(f) satisfy the same relations as in complex analysis, with regards to the coefficients an. But here another expression ψ(f) that we call cotype of f , depending on the number of zeros inside disks is very important and we show under certain wide hypothesis, that ψ(f) = ρ(f)σ(f), a formula that has no equivalent in complex analysis and suggests that it might hold in the general case. We check that ρ(f) = ρ(f), σ(f) = σ(f) and present an asymptotic relation linking the numbers of zeros inside disks for two functions of same order. That applies to a function and its derivative. We show that the derivative of a transcendental entire function f has infinitely many zeros that are not zeros of f and particularly we show that f cannot divide f when the p-adic absolute value of the number of zeros of f inside disks satisfies certain inequality and particularly when f is of finite order.

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.

We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold Cartesian product of the $p$-adic integers. Considering models in which the maximum degree that each variable appears is $N$, this expected value is \[ p^{d \lfloor \log_p N \rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for the simplest such model., 13 pages, no figures, revised to incorporate referees' comments

We present a p-adic and non-archimdean version of the Five Islands Theorem for meromorphic functions from Ahlfors' theory of covering surfaces. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed. This paper extends an earlier theorem of the author for holomorphic functions., 26 pages