Value Distribution in Ultrametric Analysis and Applications

The following sections are included:

• About the Author
• Contents
• Introduction

Affinely rigid sets have proven to play a crucial role in studying problems of uniqueness. They were introduced in [22] and were first called affinely rigid sets. Next, fearing a confusion with rigid geometry and affinoid sets, they were called stiff sets in [22]. However, the term affinely rigid was already known and popular. This is why it has become the one currently used in all further papers [3, 4, 93]. It is a very basic notion that requires to be thoroughly examined…

In this chapter, we recall basic definitions and properties on ultrametric fields: ultrametric absolute values, valuation rings, and residue fields. We must define holes of a subset and infraconnected subsets that are essential for the behavior of analytic functions (certain authors improperly call such sets “connected sets” which makes no sense in topology since there are no connected sets except singletons in an ultrametric field). A major interest of the class of infraconnected sets is that it is the biggest class of sets in an ultrametric complete algebraically closed field where the famous Krasner Mittag-Leffler theorem applies…

Monotonous and circular filters are essential on an ultrametric field, mainly because for any rational function, its absolute value admits a limit along each circular filter [36, 44, 45, 49, 53] and circular filters are the least thin filters having this property. Most of properties of analytic functions of all kinds are derived from that property of circular filters. Certain authors call “generic disk” a notion which is not clearly defined but actually represents a circular filter. We see that, given a bounded sequence, there exists a subsequence thinner than a circular filter…

Notation. As mentioned in Chapter 1, log denotes a real logarithm function of basis θ > 1 (eventually, we can take for θ an integer p that is the residue characteristic of 𝕂). When a function f from an interval I to ℝ admits a right side (respectively a left side) derivative at a point a ∈ I, we denote it by f^′r(a) (respectively f^′l(a)). If the variable is μ, we also denote it by $\frac{d_{r}f}{d\mu }$ (respectively $\frac{d_{l}f}{d\mu }$)…

The Hensel lemma is a classical tool for studying the factorization of analytic functions on a circle [1, 44] and is indispensable in Chapter 6. It is a strong result that roughly says, “In a complete field 𝕃, if $\overline{P}$ splits in the form γη with (γ, η) = $\overline{1}$, then P also splits in 𝕃[x] in the form gh with $\overline{g}=\text{γ,}\overline{h}=h,\mathrm{deg}\left(\overline{g}\right)=\mathrm{deg}\left(\text{γ}\right)$”. The proof is not very easy and requires a serious preparation. Here, we roughly follow the same process as in [1], [44]…

All considerations on analytic and meromorphic functions require considering a complete ultrametric algebraically closed field 𝕂. Here we construct the field ℂ_p and study finite extensions of ℚ_p [70]. And we show that ℂ_p is not spherically complete.

Notation. As in the previous chapters, 𝕃 denotes a complete ultrametric field whose absolute value is not trivial and whose residue class field is ℒ. We denote by F an algebraically closed ultrametric field whose absolute value is not trivial…

Notation. Recall that 𝕃 is a complete field with respect to an ultrametric absolute value. For every s ∈ ℕ*, we put $u_s=\frac{1}{p^{s-1}\left(p-1\right)}$ and r_s = p^−u_s. We study the p^sth roots of 1 and we show that they lie in circles of center 1 and radius r_s. We examine normal extensions of ℚ_p and totally ramified extensions and show the role of Eisenstein polynomials.

We need certain technical lemmas…

Several problems on p-adic analytic functions require one to consider an ultrametric algebraically closed extension of 𝕂 which is spherically complete, in order to give every circular filter a center. Others require to have a complete algebraically closed extension which admits a non-countable residue class field. Proving the existence of a spherically complete algebraically closed extension of the ground field 𝕂 isn’t easy, most of the ways involving basic considerations in logic. Here we follow the method proposed by Bertin Diarra that is only based on the notion of ultraproducts [33]…

In ℂ_p, we can define a notion of transcendence order stating that if a is transcendental over ℚ_p and has a transcendence order ≤ t and if b is transcendental over ℚ_p but algebraic over ℚ_p[a], then b also has a transcendence order ≤ t. We show the existence of numbers of order less than 1 + ϵ for every ϵ > 0 [41, 55]…

The idea of considering rational functions with no pole inside a domain D, in order to define analytic functions in D, is due to Marc Krasner [69]. The behavior of rational functions in 𝕂 is determined by circular filters which characterize all multiplicative norms on rational functions. We make a general study of the set of multiplicative semi-norms of a normed algebra which is locally compact with respect to the topology of pointwise convergence. Results are first due to Guennebaud and Garandel [61, 62]. Henceforth, the idea of considering the topologic space of multiplicative semi-norms continuous with respect to the topology of a normed algebra was used in many works on Banach algebra…

Due to the fact that any disk d(a, r) is exactly the same as d(b, r) for every b ϵ d(a, r), it is easily seen that a power series ${\displaystyle {\sum }_{n=0}^{\infty }{a}_n{\left(x-a\right)}^n}$ which admits the disk d(a, r) for disk of convergence may not be extended outside its convergence disk as it is done in complex analysis, by means of a change of origin…

Given A and B ⊂ 𝕂, f ϵ H(A) such that f(A) ⊂ B and g ϵ H(B), a basic question is whether g ° f ϵ H(A). There is an immediate application to the study of homomorphisms from an algebra H(D) to another H(D′).

Lemma 12.1. Let A and B be subsets of 𝕂 and let f ∈ H(A) be such that f(A) ⊂ B. For every λ ∈ $𝕂\backslash \overline{B}$, f − λ is invertible in H(A). Moreover, if for every h ∈ R(B), h ○ f belongs to H(A), then for every g ∈ H(B), g ○ f belongs to H(A) and for every $\lambda \in \overline{B}\backslash B$, f − λ is invertible in H(A)…

In Chapter 10, we studied and characterized the multiplicative semi-norms on a 𝕂-algebra R(D) of rational functions. We apply these properties to the completion H(D) of R(D) by considering multiplicative semi-norms that are continuous with respect to the topology of H(D). On H(D) as on R(D), the role of circular filters is obviously crucial: each continuous multiplicative semi-norm of H(D) is defined by a circular filter secant with D exactly as it was explained for rational functions. However, circular filters that are not secant with D play no role with regards to H(D)…

A power series on a p-adic field admits a disk of convergence whose radius is defined in the same way as on ℂ. The difference in behavior between power series in ℂ and in a field such as 𝕂 concerns what happens when |x| is equal to the radius of convergence. We show that the norm of uniform convergence in a disk d(a, s) ⊂ d(0, R⁻) is multiplicative and satisfies $\Vert {\displaystyle {\sum }_{n=0}^{+\infty }{a}_n{x}^n}{\Vert }_{d\left(0,s\right)}={\sup }_{n\in \mathbb{N}}\left|a_n\right|s^n$. As a consequence, the product of two power series converging in d(0, R⁻) is bounded if and only if both are bounded. We show that the algebra of power series with a radius of convergence equal to R is equal to the intersection of algebras of analytic elements H(d(0, s)) when s < R. We show that all analytic elements in d(0, R⁻) are power series converging in d(0, R⁻). The converse is false. However, we see that the analytic elements in d(0, R) are exactly the power series converging in this disk…

The wonderful Mittag-Leffler theorem for analytic elements is due to Marc Krasner who showed it on quasi-connected sets [69]. The same proof holds on infraconnected sets as it was shown by Philippe Robba [83]. The theorem shows that a Banach space H_b(D) is a direct topological sum of elementary subspaces and is indispensable to have a clear image of the space H(D). Further, it appears necessary when studying meromorphic functions as we see later…

In ℂ, it is well known that when a (not identically zero) holomorphic function admits a zero at a point α, this zero has a finite order of multiplicity. Actually, this is a generalization of a property of rational functions. In the non-Archimedean context, we find again that property among analytic elements and it is essential. In this chapter, D is just a subset of 𝕂…

We have seen that H(D) is a Banach 𝕂-algebra if and only if D is closed and bounded. But studying analytic elements, analytic functions require to know algebras of analytic elements which are not necessarily bounded. Thus, we have to examine the class Alg of subsets D of 𝕂 such that H(D) is a 𝕂-algebra with respect to usual laws [35, 37, 44, 50]…

Given an infraconnected set, the main question we consider here is whether an element f of H(D) has a derivative that belongs to H(D) and when it does, whether its Mittag-Leffler series is obtained by deriving that of f. Another question is whether an analytic element on D whose derivative is identically zero is a constant. Both questions are answered on an infraconnected closed set…

Throughout this chapter, D is infraconnected.

The function Ψ(f, μ) was defined for rational functions in Chapter 4. Here, we generalize that function to analytic elements. Its interest is to transform the multiplicative property of the norm | · | into an additive property. Overall, Ψ is piecewise affine. Long ago, such a function was first defined in classical works such as the valuation function of an analytic element [1, 35, 44] denoted by v(f, μ). However, the function v(f, μ) has the inconvenient of being contravariant: μ = − log(|x|) and v(f, −log(|x|)) = −log(|f|(r)). Here, we change both senses of variation: Ψ(f, μ) = −v(f, μ) [49, 50]…

Throughout this chapter, the set D is supposed to be infraconnected.

From Chapter 7, we know that there exists a spherically complete algebraically closed extension $\widehat{𝕂}$ of 𝕂 whose residue class field is not countable and whose valuation group is equal to ℝ. Given a subset D of 𝕂, we denote by $\widehat{D}$ the subset

Throughout this chapter, the set D is supposed to be infraconnected.

The main results given here were published in [35, 44, 50]. According to the definition of quasi-minorated elements, Theorem 21.1 is easy:

Theorem 21.1. Let f ∈ H(D). Then, f is not quasi-minorated if and only if there exists a large circular filter ℱ secant with D such that _DϕF(f) = 0…

Most of classical results on zeros of polynomials are now extended to power series. In particular, power series converging inside a disk satisfy a Schwarz lemma that is even simpler than in ℂ.

Throughout this chapter, r is a strictly positive real number and r′, r″ are strictly positive real numbers satisfying r′ < r″…

In this chapter, D is just an open subset of 𝕂.

Theorem 23.1. Let f(x) = ${\displaystyle {\sum }_{n=0}^{\infty }{a}_n{(x-a)}^n\in H(d(a,r))}$. Then the following statements (a), (b), (c), (d), and (e) are equivalent:

• (a) |a₀| > |a_n|rⁿ for all n > 1.
• (b) ||f − f(a)||_d(a,r) < |f(a)|.
• (c) f has no zero in d(a, r).
• (d) |f(x)| is constant and different from 0 in d(a, r).
• (e) f is invertible in H(d(a, r))

Throughout this chapter D is supposed to be infraconnected.

Some of the results given here were obtained in [35, 36, 44, 50]. We show that when an ideal of an algebra H(D) contains a quasi-invertible element, this ideal is principal and generated by a polynomial…

In this chapter, the field 𝕂 is supposed to have characteristic zero.

We define the p-adic logarithm and the p-adic exponential and shortly study them, in connection with the study of the roots of 1 made in Chapter 24. Both functions are also defined in [1]. Here, as in [50], we compute the radius of convergence of the p-adic exponential by using results on injectivity seen in Chapter 23…

The author is grateful to Michel Waldschmidt for his advices. On the other hand, most of results first proven in the field 𝕂 also hold (with slide changes) in an ultrametric field 𝒦 of residue characteristic 0, as, for example, the Levi-Civita field [88]…

In this chapter, we define divisors in 𝕂 or in a disk d(a, R⁻). We then define the divisor of an analytic function and of an ideal.

Definition. We call a divisor in 𝕂 (respectively a divisor in a disk d(a, R−)) a mapping T from 𝕂 (respectively from d(a, R−)) to ℕ whose support is countable and has a finite intersection with each disk d(a, r), ∀r > 0 (respectively ∀r ∈]0, R[). Thus, a divisor on 𝕂 (respectively of d(a, R−)) is characterized by a sequence (a_n, q_n)_n∈ℕ with a_n ∈ 𝕂, lim_n→∞ |a_n| = ∞ (respectively a_n ∈ d(a, R−), lim_n→∞ |a_n − a| = R), |a_n| ≤ |a_n+1|, and q_n ∈ ℕ* ∀n ∈ ℕ. So, we frequently denote a divisor by the sequence (a_n, q_n)_n∈ℕ which characterizes it…

This chapter is aimed at studying the following problem mentioned in Chapter 26 and first considered by Lazard in a tremendous work [72]. Let T be a divisor on a disk d(a, R⁻). Does there exist a function f ϵ 𝒜(d(a, R⁻)) such that 𝒟(f ) = T? The answer depends on whether or not 𝕂 is spherically complete…

In this chapter, D is a closed infraconnected set and f belongs to H(D).

The idea of factorizing semi-invertible analytic elements into a product of singular factors is a remarkable idea due to Motzkin [75]. This factorization has tight links with the Mittag-Leffler series, as shown in [36]…

Here we mean to introduce and study the notion of order of growth of an entire function on 𝕂 in relation with the distribution of zeros in disks and in relation with the question whether an entire function can be divided by its derivative inside the algebra of entire functions. First, results were published in [16, 18, 50]. The notion of order of growth was defined by Valiron [90]…

Definition and notation. In complex analysis, the type of growth is defined for an entire function having a finite order of growth t as $\sigma \left(f\right)=\lim {\sup }_{r\to +\infty }\frac{\text{Log}\left(M_f\left(r\right)\right)}{r^t}$, with t < +∞ [90]. Of course, the same notion may be defined for f ϵ 𝒜(𝕂). Here, as in Chapters 30, we denote by Log the Neperian logarithm and by e the number such that Log(e) = 1. Then, given f ϵ 𝒜*(𝕂) of order t, we set $\sigma \left(f\right)=\lim {\sup }_{r\to +\infty }\frac{\text{Log}\left(\left|f\right|\left(r\right)\right)}{r^t}$. Moreover, we put $\tilde{\sigma }\left(f\right)=\lim {\inf }_{r\to +\infty }\frac{\text{Log}\left(\left|f\right|\left(r\right)\right)}{r^t}$…

Similar to the situation in complex entire functions, here we see that the order and the type of the derivative of an entire function f are respectively equal to those of f. As in Chapters 30 and 31, we denote by Log the Neperian logarithm and by e the number such that Log(e) = 1…

In Chapters 30, 31, and 32, we defined the order of growth and the type of growth for entire functions in 𝕂 in a similar way as it is done for complex entire functions and we also defined a cotype of growth strongly linked to the order and the type: in most of the cases, the cotype is the product of the order of growth by the type of growth…

Throughout this chapter, we fix R ϵ]0, +∞ [, we denote by D the disk d(0, R⁻) and by A the 𝕂-algebra 𝒜_b(d(0, R⁻)).

In this chapter, we examine the continuous multiplicative norms and semi-norms on A. Following a work by Araujo [6], we show that there exist continuous multiplicative semi-norms whose kernel is a prime closed ideal that is neither null nor a maximal ideal. Theorem 34.1 is easily checked…

The Corona conjecture, stated by Kakutani in 1941 in the field ℂ, was solved by Carlesson in 1962 [28]. Consider the open unit disk O in ℂ and the Banach algebra B of bounded holomorphic functions in O. Each point of O obviously defines a maximal ideal of B. On the other hand, all maximal ideals are of codimension 1 and the Gelfand transform defines a topology on the maximal spectrum. The question was whether the set of maximal ideals defined by points of O was dense inside the whole spectrum of maximal ideals, with respect to the Gelfand topology…

In this chapter, we define and examine the basic properties of meromorphic functions: relations with poles of analytic elements, absolute values on fields of meromorphic functions defined by circular filters, value of the derivative on a circular filter, development in a Laurent series in an annulus, and existence of primitives [19]…

Throughout this chapter, D is infraconnected, T is a hole of D, and V is a disk of the form d(a, r) or d(a, r⁻), included in $\tilde{D}$, such that $\tilde{V}\cap D\ne \varphi$.

Definition and notation. Let f ∈ ℳ(𝕂) (respectively f ∈ ℳ(d(0, R−))) have a pole α of order q and let f(x) = ${\displaystyle {\sum }_{k=-q}^{-1}{a}_k{(x-\alpha )}^k+h(x)}$ with a_−q ≠ 0 and h ∈ ℳ(𝕂) (respectively f ∈ ℳ(d(0, R−))) and h holomorphic at α. Accordingly to previous notations for analytic elements in Chapters 11 and 15, the coefficient a₋₁ is called residue of f at α and denoted by res(f,α)…

Let us recall that the ring of analytic functions on a region of the complex number field is well known to be a Bezout ring. The two fundamental theorems necessary for a proof are the Weierstrass factorization theorem and the Mittag-Leffler theorem.

According to results of [72], it appears that in several hypotheses, rings of analytic functions on complete ultrametric algebraically closed fields are Bezout rings. However, that interesting property is not stated. Moreover, it derives from a Mittag-Leffler theorem referred in general topology whose justification is not relevant. Here we plan to give proofs of all these properties, using results on quasi-invertible analytic elements and on a Mittag-Leffler theorem for meromorphic functions similar to this of complex analysis but is quite different from Krasner’s Mittag-Leffler theorem for analytic elements on an infraconnected subset of 𝕂…

Notation. We fix R > 0 and denote by I the interval [R, + ∞[. Throughout this chapter, we denote by S the disk d(0, R⁻) and put D = 𝕂\S.

We denote by H₀(D) the 𝕂-subvector space of the f ∈ H(D) such that lim_|x|→+∞ f(x) = 0…

Given a holomorphic function f in an open bounded connected subset D of ℂ, f |(z)| reaches its maximum on the boundary of D. Consider now a closed bounded infraconnected subset D of 𝕂 and f ∈ H(D). We show that the supremum of the ø(f) when ø runs in Mult(H(D), ‖ · ‖_D) is reached on the Shilov boundary that we characterize. We show that the set of circular filters is provided with a tree structure and that the diameter is an increasing function with values in ℝ, defining distances associated with this structure. The first remarks on that tree structure are due to Motzkin [74], and it was thoroughly examined in [14] and in [49]. Here the structure is helpful to determine the Shilov boundary for algebras H(D)…

We show that every element f ∈ H(D) has continuation to a mapping f_* from Φ(D)) to Φ(𝕂). Given a circular filter ℱ ∈ Φ(D), the mapping that associates with each f ∈ H(D) the circular filter f_*(ℱ) is uniformly continuous with respect to the norm of H(D) and the metric δ on Φ(𝕂) [492]…

In this chapter, D is an infraconnected affinoid subset of 𝕂.

This chapter is aimed at characterizing the injective meromorphic functions in a subset that is a chained union of infraconnected affinoid subsets. The relation satisfied by such injective functions recalls the one obtained in [58] by Yvette Perrin. The equality

was shown for various kinds of injective analytic functions. In Chapter 49 of [44], it was suggested that all injective analytic elements should satisfy that by making the conjecture that an injective analytic element should be the product of a major Moebius function by a train of minor Moebius functions…

The Nevanlinna theory was made by Rolf Nevanlinna on complex functions [79] and widely used by many specialists of complex functions, particularly Walter Hyman [65]. It consists of defining counting functions of zeros and poles of a meromorphic function f and giving an upper bound for multiple zeros and poles of various functions f − b, b ϵ ℂ…

In Chapter 31, we noted the following question:

Let $f\left(x\right)={\displaystyle {\sum }_{n=0}^{\infty }{b}_n{x}^n\in 𝒜\left(𝕂\right)}$ be such that 0 < ρ(f) < + ∞. Do we have

Actually, the answer is no in the general case. Using the counting function of an entire function, we construct a counter-example. In order to prove that, we need two basic lemmas…

We can now prove the second main theorem under different forms. Lemma 45.1 is essential and directly leads to the theorems.

Lemma 45.1. Let f ∈ ℳ(𝕂) (respectively f ∈ℳ_u(d(0, R−))). Suppose that there exists ξ ∈ 𝕂 (respectively ξ ∈ ℳ_b(d(0, R−))) and a sequence of intervals I_n = [u_n, v_n] such that u_n < v_n < u_n+1, lim_n→+∞ u_n = +∞ (respectively lim_n→+∞ u_n = R) and

(respectively lim_n→+∞ (inf_r∈In T(r, f) − Z(r, f − ξ)) = +∞)…

Throughout Chapters 46–57, the field 𝕂 is supposed to be of characteristic 0.

Now, we mean to construct a Nevanlinna theory for meromorphic functions in the complement of an open disk thanks to the use of specific properties of the analytic elements on infraconnected subsets of 𝕂 already examined…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

Notation. In all the chapters, the field 𝕂 is supposed to be of characteristic 0. As in Chapter 41, we denote by D the set 𝕂\d(0, R−) with R a positive number. The definitions of 𝒜(D), 𝒜^c(D), ℳ(D), and ℳ^c(D) are those given in Chapter 46…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

We examine particular cases where curves are defined by their equations so that, for most of them, the p-adic Nevanlinna theory lets us find easy proofs. Most of results come from [24]…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

In complex functions theory, a notion closely linked to Picard’s exceptional values was introduced: the notion of “perfectly branched value” [29]. Here we consider the same notion on ℳ(𝕂), on ℳ(d(a, R⁻)), and on ℳ(D). Most of results come from [51]…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

This chapter is aimed at studying various properties of derivatives of meromorphic functions, particularly their sets of zeros. Many important results are due to Jean-Paul Bézivin [10, 11, 12].

We first note a general property concerning quasi-exceptional values of meromorphic functions and derivatives…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

Small functions with respect to a meromorphic function are well known in the general theory of complex functions. Particularly, one knows the Nevanlinna theorem on three small functions. Here we construct a similar theory…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

Definition. Two functions f, g defined in a set E are said to share a function h, ignoring multiplicity if f(x) – h(x) = 0 is equivalent to g(x) – h(x) = 0.

This kind of problem was considered in [95]…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

In the 1950s, Walter Hayman asked the question of whether, given a meromorphic function in ℂ, the function g′gⁿ might admit a quasi-exceptional value b ≠ 0 [66]. Hayman showed that g′gⁿ has no quasi-exceptional value, whenever n ≥ 3. Henceforth, the problem was solved for n = 2 by Mues in 1979 [77] and next for n ≥ 1, in 1995 by Bergweiler and Eremenko [8] and separately by Chen and Fang [30]. See also [95]. The same problem is posed on the field 𝕂, both in ℳ(𝕂) and in a field ℳ(d(a, R⁻)) (a ϵ 𝕂, R > 0). These are the studies in [52, 80]…

Throughout this chapter, the field 𝕂 is supposed to be of characteristic 0.

In order to look for meromorphic functions of uniqueness in the following chapter, here we first examine the composition of meromorphic functions which, in the general case, is not a meromorphic function.

Throughout this chapter, 𝔼 is an algebraically closed field of characteristic 0 without any assumption on the existence of an absolute value. The field 𝕂 is supposed to have characteristic zero…

Throughout this chapter, 𝕂 is of characteristic 0.

We want to study sufficient conditions on a meromorphic function h assuring that if the composition of meromorphic functions of the form h ∘ f and h ∘ g are equal, then f and g are equal. This kind of problem follows many other problems of uniqueness studied, particularly on unique range sets with (or without) multiplicities and polynomials of uniqueness for analytic or meromorphic functions in the complex field and in an ultrametric field [3, 47, 93, 98]. Polynomials of uniqueness were introduced and studied in ℂ and in a p-adic field by Yi and Yang [98], Fujimoto [60], and Li and Yang [97]. Here most results come from [47]…

Throughout this chapter, 𝕂 is of characteristic 0.

We introduce urscm and ursim for p-adic meromorphic functions. Many studies were made in the eighties and the nineties concerning functions in ℂ, [7, 59, 77, 78]. Studies were also made in the non-Archimedean context since the late nineties [3, 4, 21, 22, 27].

Here, we only consider the situation in an ultrametric field…

Throughout this chapter, 𝕂 is of characteristic 0.

We denote by 𝔼 an algebraically closed field of characteristic 0. The field 𝕂 is supposed to have characteristic 0.

In Chapter 56, we have constructed sets of n points which are ursims for p-adic entire functions whenever n ≥ 9 and are ursims for p-adic meromorphic functions whenever n ≥ 16. Here, we construct new sets which are ursims of n points for 𝔼[x] and for 𝒜(𝕂) for every n ≥ 9 but then, we show that such sets are never urscm and of course are not ursim for ℳ(𝕂) and for 𝔼(x), although they are not preserved by any Moebius functions, a contradiction to a natural expectation…

Results in characteristic p were published in [25, 26, 27].

Notation. In this chapter, we denote by p the characteristic of 𝕂 and by q its characteristic exponent, i.e. q = p if p ≠ 0, and q = 1 if p = 0.

As usual, given a ∈ 𝕂 and n ∈ ℕ, we denote by $\sqrt[p^n]{a}$ the unique b ∈ 𝕂 such that b^(pn) = a…

Notation. Throughout Chapter 54, 𝕂 is a field of characteristic p ≠ 0, α belongs to 𝕂, and R belongs to ]0, +∞[. We try to generalize the results of Chapters 52 and 53 when the characteristic is p. Many results come from [22, 23].

Lemma 59.1. Let P(x) = xⁿ − x^m + k with m < n and k ≠ 0. Let f, g ∈ 𝒜(𝕂)\𝕂 (respectively f, g ∈ 𝒜_u(d(α, R−))) satisfy P(f) = λP(g) with λ ∈ 𝕂*. Then f and g have the same ramification index…

Throughout this chapter, the field 𝕂 is supposed to have characteristic p ≥ 0 and characteristic exponent q. R is a strictly positive number.

Most of results come from [46, 64].

In the sequel, we use the following basic lemmas:

Lemma 60.1. Let 𝔼 be a field, and let P ∈ 𝔼[x] satisfy deg(P) = 3 and be such that P′ has two distinct zeros c₁, c₂. Then, P(c₁) ≠ P(c₂)…

Throughout this chapter, the field 𝕂 has characteristic p ≥ 0.

We call Yoshida’s equation a differential equation of the form (ε) (y′)^m = F (x, y) (with F(x, y) ϵ 𝕂 (x, y\𝕂). Several results were obtained in characteristic 0 and in characteristic p ≠ 0, for meromorphic functions in the whole field 𝕂 or inside a disk d(a, R⁻). In [24], it was shown that if (ε) admits solutions in ℳ(𝕂)\𝕂(x), then F ϵ 𝕂(x)[y] and deg_y (F) ≤ 2m. Moreover, it was shown that if F ϵ 𝕂(y), then any solution of the equation lying in ℳ(𝕂) is a rational function with a very specific form. That was generalized in characteristic p ≠ 0 in [27]. Theorems 61.1 and 61.2 were proven when the ground field is ℂ_p. They have an immediate generalization in any algebraically closed field complete for an ultrametric absolute value, such as 𝕂…

Here we want to consider Yoshida’s equation for meromorphic functions inside a disk d(a, R⁻) [23]. Several methods look like parts of the proof of Theorem 61.3. However, we do not obtain a result as general as in Chapter 61.

Throughout this chapter, 𝕂 has characteristic 0…

The following sections are included:

• Bibliography
• Definitions
• Notations