An ultrametric Nevanlinna’s second main theorem for small functions of a special type

In ultrametric Nevanlinna theory, the Nevanlinna’s second main theorem for small functions has only been proved in the case of at most three small functions. In this paper, we prove a second main theorem for q small functions of a special type when the residue characteristic of the field is zero. Le theoréme de Nevanlinna ultramétrique pour petites fonctions Résumé En théorie de Nevanlinna ultramétrique, le second théorème fondamental de Nevanlinna pour des petites fonctions a seulement été établi pour trois petites fonctions. Dans cet article, on montre un second théorème fondamental pour q petites fonctions d’un certain type quand la caractéristique résiduelle du corps est zero.


Introduction
The Nevanlinna Second Main Theorem on three small functions is well known in complex meromorphic functions.It is easily proved by using the so-called bi-ratio technique.However, there is no way to generalize such a technique to n small functions.In 2004, Yamanoi proved the Second Main Theorem on n small functions in [6] by a method that has no link with the elementary proof of the theorem on three small functions.Now, let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value.The Nevanlinna Second Main Theorem does exist for meromorphic functions in the whole field and also for meromorphic functions in an open disk, see [1], [2], [3].In the same way as for complex functions, an ultrametric version of Nevanlinna Second Main Theorem on three small functions can be proved by the technique of bi-ratio.But it is impossible to adapt to p-adic analysis Yamanoi's work that mainly uses typically archimedean properties in the complex field, particularly integrations on paths.Therefore it remains open how to obtain a similar result in an ultrametric field and also, whether it can be obtained.This is why, here, we will try to obtain a kind of Nevanlinna Second Main Theorem on n small functions in ultrametric analysis, in a case where such a solution appears: we place ourselves in a field of residue characteristic zero, consider just analytic functions inside an open disk and assume that the "small functions" actually are bounded and satisfy an additional condition: intersections of two global images are empty.The hypothesis of zero residue characteristic is due to the behaviour of derivatives in a field of residue characteristic p = 0, which is explained after the proof of the theorem.
Let K be an algebraically closed field of characteristic zero and of residue characteristic zero, complete for an ultrametric absolute value For each r ∈]0, R[, let η(r, f ) denote the number of zeros of f in the circle C(0, r) counting multiplicity, and let η(r, f ) be the number of zeros of f in C(0, r) ignoring multiplicity. Let . Let (a n ) n∈N * be the sequence of zeros of f with 0 < |a n | ≤ |a n+1 |, with a 0 whenever f (0) = 0 and let k n denote the order of the zero a n .Moreover, if f (0) = 0, we denote by k 0 the order of the origine as a zero and we set l 0 = 1.And if f (0) = 0 we set k 0 = l 0 = 0.
Then we define the counting function of zeros of f as Respectively, let the counting function ignoring multiplicities be defined as Z(r, f ) = l 0 log r + |an|≤r (log r − log |a n |).
Remark 1.2.As far as analytic functions are involved, it is now easy to state the second main theorem by only using the function Z(r, f ), this function being equivalent to the characteristic function T (r, f ) for analytic functions.

Definition 1.3. We call a function
be of the form ω j = a j + θ j , j = 1, . . ., q, where a j ∈ K and Then Proof.Let ω j = a j + θ j , 1 ≤ j ≤ q and let Since φ α,r (f ) is an increasing function of r, and tends to infinity as r tends to R, there exists a unique radius ρ α,j for which By hypothesis (2.2), it follows that f − ω k has no zeros in d(α, ρ − α,j ) for k = j.On the other hand, since φ α,ρ α,j (f − a j ) > ||θ j ||, the number of zeros of f − a j in d(α, ρ − α,j ) is equal to this of f − ω j , see Lemma 2.3.Now consider the circle C(0, r) for fixed r, and consider the possibly several points α m,j ∈ C(0, r), 1 ≤ m ≤ u j , 1 ≤ j ≤ q, such that f (α m,j ) = ω j (α m,j ).For each indices m and j, let σ m,j denote the number ρ α m,j ,j previously defined as ρ α,j for α.Now the disks d(α m,j , σ − m,j ) are pairwise disjoint.For every m = 1, . . ., u j and j = 1, . . ., q, let ν m,j be the number of zeros of f − ω j in d(α m,j , σ − m,j ) counting multiplicities, and let ν m,j be this number ignoring multiplicities.Hence ν m,j is also the number of zeros of f − a j in d(α m,j , σ − m,j ) counting multiplicities.Since K has residue characteristic zero, it follows from the above that the number of zeros of f in each disk d(α m,j , σ − m,j ) is exactly ν m,j − 1.
a property does not hold when the residue characteristic is p = 0.In the proof of the theorem, the equality is crucial.This is why we presently can't generalize the theorem to fields with a non-zero residue characteristic.
On the other hand, we have used the fact that |f |(r) is an increasing property for analytic functions inside a disk.This is no longer true for meromorphic functions.Therefore, generalizing the theorem to meromorphic functions does not seem possible in this way.