Value Distribution in p-adic Analysis

The following sections are included:

• Introduction
• Contents

This first chapter is aimed at recalling basic definitions and properties on ultrametric fields. We will examine ultrametric absolute values, valuation rings and residue fields. Particularly, we will define holes of a subset and infraconnected sets that are essential with regards to the behaviour 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 (among others) is that this is the biggest class of sets of an ultrametric complete algebraically closed field where the famous Krasner Mittag-Leffler theorem…

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 and circular filters are the least thin filters having this property. Most of properties of analytic functions of all kinds derive 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 will 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 (resp. a left side) derivative at a point a ∈ I, we will denote it by f′^r(a) (resp. f′^l(a)). If the variable is µ, we will also denote it by (resp. )…

The Hensel Lemma is a classical tool for studying the factorization of analytic functions on a circle and is indispensable in Chapter 5. It is a strong result that roughly says: “In a complete field L, if splits in the form γη with , then P also splits in L[x] in the form gh with , , ". The proof is not very easy and requires a serious preparation. Here we will roughly follow the same process as in [2], [63], [101] and more precisely in [69], with a few corrections…

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

Notation: For every s ∈ ℕ*, we put and r_s = p^−u_s. We will study the p^s-th roots of 1 and we will show that they lie in circles of center 1 and radius r_s. We will examine normal extensions of ℚ_p and totally ramified extensions and show the role of Eisenstein polynomials…

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 is not easy, most of the ways involving basic considerations in logic. Here we will follow the method proposed by Bertin Diarra in [66], that is only based on the notion of ultraproducts…

The transcendence order of a number a in ℂ_p was introduced in [58]. This notion, which is specific to p-adic numbers, looks a bit like that of transcendence type but it is quite different because this concerns transcendence over ℚ_p, not over ℚ. In 1978 the notion of transcendence order was defined and we showed the existence of numbers with transcendence order ≤ 1 + ϵ. B. Guralnick suggested me a shorter proof of Theorem 8.6 that we present here with more details, thanks to Bertin Diarra. This states that if a is transcendental over ℚ_p and has a transcendence order ≤ t and if b is trenscendental over ℚ_p but algebraic over ℚ_p[a], then b also has a transcendence order ≤ t. Finally, in Theorem 8.7 we construct numbers with an infinite order of transcendence. In [58] the existence of p-adic numbers of order ≤ 1 was stated but the proof contained a mistake that puts in doubt the result. This is why we will only show the existence of numbers of order less than 1 + ϵ for every ϵ > 0…

The transcendence type is defined in ℂ_p in the same way as in ℂ.

Definitions and notation: Given a complex number z, we denote by |z|_∞ its modulus. Throughout this chapter, a number a ∈ ℂ_p will just be said to be algebraic (resp. transcendental) if it is algebraic (resp. transcendental) over ℚ. When a is algebraic or transcendental over ℚ_p we will precise this. We will denote by Ω the field of algebraic numbers and by A the ring of algebraic integers…

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. The behaviour of rational functions in 𝕂 is determined by circular filters which characterize all multiplicative norms on rational functions. We will make a general study of the set of multiplicative semi-norms of a normed algebra which is is locally compact with respect to the topology of pointwise convergence. Results are first due to B. Guennebaud and G. Garandel…

Due to the fact that any disk d(a, r) is exactly the same as d(b, r) for every b ∈ d(a, r), it is easily seen that a power series 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 ∈ H(A) such that f(A) ⊂ B and g ∈ H(B), a basic question is whether g ○ f ∈ H(A). There is an immediate application to the study of homomorphisms from an algebra H(D) to another H(D′)…

In Chapter 10 we studied and characterized the multiplicative semi-norms on a 𝕂-algebra R(D) of rational functions. We will 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 of behaviour 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 . 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 will 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. The same proof holds on infraconnected sets as it was shown by Philippe Robba. 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 will 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…

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 clopen set. Throughout this chapter D is a subset of 𝕂 and is supposed to be open and infraconnected and we fix R > 0…

Throughout this chapter D is infraconnected.

The function Ψ(f, μ) was defined for rational functions in Chapter 3. Here we will generalize that function to anlaytic 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 analyitc element 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 will change both senses of variation: Ψ(f, μ) = −v(f, −μ)…

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

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

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

The main results given here were published in [51], [53], [63]. 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 _Dφ_ℱ(f) = 0.

Proof. By Lemmas 17.12 and 16.7, without loss of generality we can assume that D is bounded…

Most of classical results on zeros of polynomials will now be 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″…

Throughout this chapter D is supposed to be infraconnected and closed.

We consider the question whether every element of H_b(D) reaches its maximum ‖f‖_D at some point α ∈ D.

Definition: D is said to be strongly infraconnected if for every hole T = d(a, r⁻) with r ∈ |L|, there exists a sequence (x_n)_n∈ℕ in D such that |x_n − a| = |x_n − x_m| = r whenever n ≠ m…

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

Theorem 24.1: Let . Then the following statements a), b), c), d), e) are equivalent:

• |a₀| > |a_n|rⁿ for all n > 1,
• ‖f − f(a)‖_{d(a, r)} < |f(a)|,
• f has no zero in d(a, r),
• |f(x)| is constant and different from 0 in d(a, r),
• f is invertible in H(d(a, r))…

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

We will define the p-adic logarithm and the p-adic exponential and will shortly study them, in connection with the study of the roots of 1 made in Chapter 6. Both functions are also defined in [2]. Here, as in [61], we compute the radius of convergence of the p-adic exponential by using results on injectivity seen in Chapter 24…

The six exponential problem is well known on ℂ and was solved by Serge Lang and K. Ramachandra. The problem is the following: let a₁, a₂, a₃, (resp. b₁, b₂ ∈ ℂ) be ℚ-linearly independant. Then at least one of the six numbers e^a_ib_j is transcendental. Next, consider the same problem with only four exponentials: let a₁, a₂, (resp. b₁, b₂ ∈ ℂ) be ℚ-linearly independant. The question is whether one of the numbers e^a_ib_j is transcendental: this is the four exponentials conjecture on ℂ due to Serge Lang…

Throughout this chapter D is supposed to be infraconnected.

Some of the results given here were obtained in [51] and published in [53] and in [63]. We will 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 paragraph we shall define divisors in 𝕂 or in a disk d(a, R⁻). We then shall define the divisor of an analytic function and of an ideal. Given a divisor T on 𝕂, there is no problem to construct an entire function whose divisor is T. But given a divisor T on a disk d(a, r⁻), it is not always possible to find an analytic function (in that disk) whose divisor is T. This is Lazard's problem that we will examine in the next chapter…

This chapter is aimed at studying the following problem mentioned in Chapter 28 and first considered by M. Lazard in a tremendous work. Let T be a divisor on a disk d(a, R⁻). Does exist a function f ∈ 𝒜(d(a, R⁻)) such that 𝒟(f) = T? The answer depends on whether or not 𝕂 is spherically complete…

The concept of sets of range uniqueness (SRU) was introduced by Diamond, Pomerance and Rubel for complex analytic functions. It is a generalization of the Identity Theorem. Several other papers on this topic appeared all over the last 20 years. However, few took place in a p-adic field: the results presented here comes from [21]. Others concerning analytic elements are given in [22]…

Throughout 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 E. Motzkin. This factorization has tight links with the Mittag-Leffler series, as it was shown in [61], [62]. However, this factorization requires that the set D belongs to Alg…

In this chapter, we will 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, existence of primitives…

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 , such that .

Definition and notation: Let f ∈ ℳ(𝕂) (resp. f ∈ ℳ(d(0, R⁻)) have a pole α of order q and let with a_−q ≠ 0 and h ∈ ℳ(𝕂) (resp. 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)…

In complex analysis, studies were made on the identity theorem for meromorphic functions of bounded characteristic in the unit disk. Let (z_n)_n∈ℕ be a sequence of distinct points in the unit disk D = {z ∈ ℂ | |z| < 1} such that and . W. K. Hayman showed that if f is a meromorphic function of bounded characteristic in D satisfying f(z_n) = 0 for every n ∈ ℕ, then f is identically zero in D…

Throughout the chapter, we fix R ∈]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 will examine the continuous multiplicative norms on A and, following a recent work by J. Araujo, we will show that there exist continuous multiplicative semi-norms whose kernel is a prime closed ideal that is neither null nor a maximal ideal. Theorem 35.1 is easily checked…

The Corona conjecture, stated by Kakutani in 1941 in the field ℂ, was solved by L. Carlesson in 1962. 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…

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 will show that the supremum of the ϕ(f) when ϕ runs in Mult(H(D), ‖ ⋅ ‖_D) is reached on the Shilov boundary that we will characterize. We will 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 to this structure. The first remarks on that tree structure are due to E. Motzkin, and it was thoroughly examined in [16]. Here the structure will be 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 to 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)…

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

The 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 [82] by Yvette Perrin. The equality

The Nevanlinna Theory was made by Rolf Nevanlinna on complex functions. 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 ∈ ℂ…

Throughout the chapter, the field 𝕂 is supposed to have characteristic 0.

As immediate applications of the Second Main Theorem, we can notice the following Theorems 41.1, 41.2, 41.3, 41.4.

Theorem 41.1: Let a₁, a₂ ∈ 𝕂 (a₁ ≠ a₂) and let f, g ∈ 𝒜(𝕂) satisfy f⁻¹({a_i}) = g⁻¹({a_i}) (i = 1, 2). Then f = g.

Theorem 41.2: Let a₁, a₂, a₃ ∈ 𝕂 (a_i ≠ a_j ∀i ≠ j) and let f, g ∈ 𝒜_u(d(a, R⁻)) satisfy f⁻¹({a_i}) = g⁻¹({a_i}) (i = 1, 2, 3). Then f = g.

Theorem 41.3: Let a₁, a₂, a₃, a₄ ∈ 𝕂 (a_i ≠ a_j ∀_i ≠ j) and let f, g ∈ ℳ(𝕂) satisfy f⁻¹({a_i}) = g⁻¹({a_i}) (i = 1, 2, 3, 4). Then f = g.

Theorem 41.4: Let a₁, a₂, a₃, a₄, a₅ ∈ 𝕂 (a_i ≠ a_j ∀_i ≠ j) and let f, g ∈ ℳ_u(d(a, R⁻))) satisfy f⁻¹({a_i}) = g⁻¹({a_i}) (i = 1, 2, 3, 4, 5). Then f = g…

We will 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 [37]…

Small functions with respect to a meromorphic functions are well known in the general theory of complex functions. Particularly, one knows the Nevanlinna theorem on 3 small functions. Here we will construct a similar theory…

The 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 Bezivin…

In the fifties, Walter Hayman asked the question whether, given a meromorphic function in ℂ, the function g′gⁿ might admit a quasi-exceptional value b ≠ 0. W. Hayman showed that g′gⁿ has no quasi-exceptional value, whenever n ≥ 3. Henceforth, the problem was solved for n = 2 by E. Mues in 1979 and next for n ≥ 1, in 1995 by W. Bergweiler and A. Eremenko and separately by H. Chen and M. Fang. The same problem is posed on the field 𝕂, both in ℳ(𝕂) and in a field ℳ(d(a, R⁻)) (a ∈ 𝕂, R > 0)…

Throughout the chapter the field 𝕂 is supposed to have characteristic 0.

In studying meromorphic functions, several problems come from the residue characteristic p when p ≠ 0. Optimal functions that we are going to define can avoid problems due to the residue characteristic…

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. Results were published in [25]…

Definition and notation: In complex analysis, the type of growth is defined for an entire function of order t as , with t < +∞. Of course the same notion may be defined for f ∈ 𝒜(𝕂). Here, as in Chapters 47 and 49, we put θ = e and we denote by log the Neperian logarithm. Then, given f ∈ 𝒜⁰(𝕂) of order t,

Similarly to the situation in complex entire functions, here we will 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 47 and 48, we put θ = e and denote by log the Neperian logarithm…

In complex functions theory, a notion closely linked to Picard's exceptional values was introduced: the notion of “perfectly branched value”. Here we shall consider the same notion on ℳ(𝕂) and on ℳ(d(a, R⁻)). Most of results come from [75], [79] and [33]…

Affinely rigid sets have proven to play a crucial role in studying problems of uniqueness. They were introduced in [35] and were first called affinely rigid sets. Next, fearing a confusion with rigid geometry and affinoid sets, they were called stiff sets in [35]. However, the term affinely rigid was already known and popular. This is why it has become the one currently used in all further papers [4], [5], [6], [7], [8], [136]. It is a very basic notion that requires to be thoroughly examined…

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

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. Polynomials of uniqueness were introduced and studied in ℂ and in a p-adic field by X.H. Hua and C.C. Yang, H. Fujimoto, P. Li and C.C. Yang, H. Khoai and C.C. Yang. Similarly, E. Mayerhofer and the author made a study of a rational function of uniqueness. In Chapter 41 we defined polynomials of uniqueness; here this notion will be generalized. Similarly, we will consider the same question in the purely algebraic context. All results come from [71]…

We shall introduce urscm and ursim for p-adic meromorphic functions. Many studies were made in the eighties and the nineties concerning functions in ℂ. Studies were also made in the non-archimedean context since the late nineties…

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

In Chapter 54 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 will construct new sets which are ursims of n points for 𝔼[x] and for 𝒜(𝕂) for every n ≥ 9 but then, we will 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…

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

Many problems were considered on ℂ as on a p-adic field, involving sharing values for functions of the form f′P′(f). Here we consider similar problems. They are deeply linked to polynomials of uniqueness. First we will make some general considerations. Next, we will examine the following question: given two meromorphic functions f, g and a polynomial P, can we find sufficient conditions assuring that f′P′(f)g′P′(g) is a small function with respect to f and g? The solution of this problem is indispensable when we consider value sharing problems for functions f′P′(f), g′P′(g). Theorem 56.3 was published in [29] and the same ones for complex meromorphic functions in [30]…

The problems we will consider was posed and examined first on ℂ and received various solutions by taking particular forms for a polynomial P. Here we place ourselves on the p-adic field 𝕂 and we will give the polynomial P a form as general as possible in the two next chapters. Here we will prove technical lemmas that are indispensable in Chapter 59…

Value sharing a small function (CM) has been a classical topic for complex and p-adic entire or meromorphic functions. In this chapter, we will first consider the problem for of p-adic analytic functions…

After examining the problem of value sharing for analytic functions, we will now consider meromorphic functions. All results come from [29]. The method is inspired from [103] but the set of polynomials we consider is wider and there is no restriction on the shared small function. A similar method lets us obtain analogue results for complex meromorphic functions…

Results in characteristic p were published in [38].

Notations: 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 the unique b ∈ 𝕂 such that b^(pn) = a…

Notation: Throughout Chapter 61, 𝕂 is a field of characteristic p ≠ 0, α belongs to 𝕂 and R belongs to ]0, +∞[. We will try to generalize results of Chapters 52, 53 and 54 when the characteristic is p. Many results come from [38] and [39]…

Throughout the 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 [67]. Certain similar problems were examined in a p-adic field in [90] and in ℂ in [139]…

In this chapter, we try to replace a decomposition involving polynomials by one involving rational functions. Most of results were given in [104].

Notation: Throughout the chapter, we suppose that 𝕂 has characteristic 0 and 𝔼 is a field of characteristic 0 with no hypothesis of absolute value.

We shall consider F, G ∈ 𝔼(x), (resp. F, G ∈ 𝕂(x)) , with A, B, C, D ∈ 𝔼[x] (resp. A, B, C, D ∈ 𝕂[x]) and gcd(A, B) = gcd(C, D) = 1…

Throughout the chapter, the field 𝕂 has characteristic p that is either equal to 0 or to a prime number.

We call Yosida's Equation a differential equation of the form (Ɛ) (y′)^m = F(x, y) (with F(x, y) ∈ 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 [37], it was shown that if (Ɛ) admits solutions in ℳ(𝕂)\𝕂(x), then F ∈ 𝕂(x)[y] and deg_y(F) ≤ 2m. Moreover, it was shown that if F ∈ 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 [38]. Theorems 64.1 and 64.2, were proven in [34] when the ground field is ℂ_p…

Here we want to consider Yosida's Equation for meromorphic functions inside a disk d(a, R⁻). Several methods look like parts of the proof of Theorem 64.3. However we do not obtain a result as general as in Chapter 64…

The following sections are included:

• Bibliography
• Definitions
• Notations