# A survey on a few recent papers in p-adic Value Distribution

Alain Escassut

► To cite this version:

Alain Escassut. A survey on a few recent papers in p-adic Value Distribution. J.M.M.S., 2022, 1 (4), pp.27-56. hal-04201250

HAL Id: hal-04201250

https://uca.hal.science/hal-04201250v1

Submitted on 9 Sep 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# A survey on a few recent papers in p-adic Value Distribution

by Alain Escassut

*In memory of Professor Wolfgang Tutschke*

In this article, we propose to present several recent results: a new proof of the p-adic Hermite-Lindemann Theorem, a new proof of the p-adic Gel'fond-Schneider Theorem, exceptional values of meromorphic functions and derivatives and the p-adic Nevanlinna theory applied to small functions. We first have to recall the definitions of the p-adic logarithm and exponential.

## 1 Logarithm and exponential in a p-adic field

**Notations:** We denote by $\mathbb{Q}_p$ the completion of $\mathbb{Q}$ with respect to the p-adic absolute value and by $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$, which is known to be algebraically closed [7]. In general, we denote by $\mathbb{K}$ an algebraically closed field of characteristic 0 complete with respect to an ultrametric absolute value, such as $\mathbb{C}_p$. The ultrametric absolute value of $\mathbb{K}$ is denoted $|\cdot|_{\infty}$ while the archimedean absolute value of $\mathbb{C}$ is denoted $|\cdot|_{\infty}$.

Let $a \in \mathbb{K}$ and let $R \in \mathbb{R}_+$. We denote by $d(a, R)$ the "closed" disk $\{x \in \mathbb{K} \mid |x-a| \le R\}$ and by $d(a, R^{-})$ the "open" disk $\{x \in \mathbb{K} \mid |x-a| < R\}$.

We denote by $\mathcal{A}(\mathbb{K})$ the algebra of power series converging in all $\mathbb{K}$. Given $a \in \mathbb{K}$ and $R > 0$, we denote by $\mathcal{A}(d(a, R^{-}))$ the algebra of power series $\sum_{j=0}^{\infty} a_n (x-a)^n$ converging in $d(a, R^{-})$ and by $\mathcal{A}_b(d(a, R^{-}))$ the subalgebra of functions $f(x) \in \mathcal{A}_b(d(a, R^{-}))$ that are bounded in $(d(a, R^{-}))$ and we put $\mathcal{A}_u(d(a, R^{-})) = \mathcal{A}(d(a, R^{-})) \setminus \mathcal{A}_b(d(a, R^{-}))$.

Moreover we denote by $H(d(a, r))$ the algebra of power series $\sum_{j=0}^{\infty} a_n (x-a)^n$ converging in $d(a, R)$ called *analytic elements in d(a, R)*. Given an element *f* of $H(d(0, R))$ we put $|f|(r) = \sup_{x \in d(0, R)} |f(x)|$.

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. Here, as in [7], we compute the radius of convergence of the *p*-adic exponential by using results on injectivity.

The following lemma 1.a is easy:

**Lemma 1.a:** $\mathbb{K}$ is supposed to have residue characteristic $p \neq 0$. Let $r \in ]0, 1[$ and for each $n \in \mathbb{N}$, let $h_n(x) = (1+x)^{p^n}$. The sequence $h_n$ converges to 1 with respect to the uniform convergence on $d(0, r)$.
**Notations:** We denote by log the real logarithm function of base e. Given a power series $\sum_{j=0}^{\infty} a_j x^j$ converging in $d(0, R^{-})$ and given a number $\mu < \log(R)$ we denote by $\nu^+(f, \mu)$ the biggest integer $q$ such that $\sup_{j \ge 0} \log(|a_j|) + j\mu = \log(|a_q|) + q\mu$.

For each $q \in \mathbb{N}^*$ we denote by $R_q$ the positive number such that $\log_p(R_q) = -\frac{1}{p^{q-1}(p-1)}$. We denote by $g(x)$ the series $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}$.

The following lemma 1.b is well known (Theorem B.13.7 in [7]):

**Lemma 1.b:** Let $f(x) = \sum_{j=0}^{\infty} a_j x^j$ be converging in $d(0, R^{-})$ and let $r < R$.

Then $\nu^+(f, \log(r))$ is the number of zeros of $f$ in $d(0, r)$, taken multiplicity into account.

**Theorem 1.1:** $g$ has a radius of convergence equal to 1. If the residue characteristic of $K$ is $p \neq 0$, then $g$ is unbounded in $d(0, 1^{-})$. If the residue characteristic is zero, then $|g(x)|$ is bounded by 1 in $d(0, 1^{-})$. The function defined in $d(1, 1^{-})$ as $\text{Log}(x) = g(x-1)$ has a derivative equal to $\frac{1}{x}$ and satisfies $\text{Log}(ab) = \text{Log}(a) + \text{Log}(b)$ whenever $a, b \in d(1, 1^{-})$.

*Proof.* It is clearly seen that the radius of $g$ is 1, because $|n| \ge \frac{1}{n}$ and $|n| \le 1$ for all $n \in \mathbb{N}$. As in the Archimedean context, the property $\text{Log}(ab) = \text{Log}(a) + \text{Log}(b)$ comes from the fact that both $\text{Log}$ and the function $h_a$ defined as $h_a(x) = \text{Log}(ax)$ have the same derivative. The other statements are immediate. $\square$

**Notation:** When $K$ has residue characteristic $p \neq 0$, we introduce the group $W$ of the $p^s$-th roots of 1, i.e., the set of the $u \in K$ satisfying $u^{p^s} = 1$ for some $s \in \mathbb{N}$.

Recall that analytic elements were defined by M. Krasner and are defined in [7].

**Theorem 1.2:** $K$ is supposed to have residue characteristic $p \neq 0$ (resp. 0). All zeros of $\text{Log}$ are of order 1. The set of zeros of the function $\text{Log}$ is equal to $W$, (resp. 1 is the only zero of $\text{Log}$). The restriction of $\text{Log}$ to the disk $d(1, (R_1)^{-})$ (resp. $d(1, 1^{-})$) is injective and is a bijection from $d(1, (R_1)^{-})$ onto $d(0, (R_1)^{-})$ (resp. from $d(1, 1^{-})$ onto $d(0, 1^{-})$).

*Proof.* It is obvious that the zeros of $\text{Log}$ are of order 1 because the derivative of $\text{Log}$ has no zero. First, we suppose $K$ to have residue characteristic $p \neq 0$. Each root of 1 in $d(1, 1^{-})$ is a zero of $\text{Log}$. Moreover, by Theorem A.6.8 of [7], we know that the only roots of 1 in $d(1, 1^{-})$ are the $p^n$-th roots. Now we can check that $\text{Log}$ admits no zero other than the roots of 1. Indeed, suppose that $a$ is a zero of $\text{Log}$ but is not a root of 1, and for each $n \in \mathbb{N}$, let $b_n = a^{p^n}$.
Since $b_n$ belongs to $d(1, 1^{-})$, by Lemma B.16.1 of [7] we have $\lim_{n \to \infty} b_n = 1$. But obviously $\mathrm{Log}(b_n) = 0$ for every $n \in \mathbb{N}$, hence this contradicts the fact that 1 is an isolated zero of $\mathrm{Log}$.

Thus, $\mathrm{Log}$ has no zero in the disk $d(1, (R_1)^{-})$, except 1 and therefore, by Lemma 1.b the series $f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}$ satisfies $\nu^+(f, \log r) = 1$ for every $r \in ]0, R_1[$, hence $r > \frac{r^n}{|n|}$ for all $r \in ]0, R_1[$, for every $n \in \mathbb{N}^*$. Therefore, by Corollary B.14.10 of [7] it is injective in $d(0, R_1^{-})$. Then, by Corollary B.13.10 of [7], we see that $\mathrm{Log}(d(1, R_1^{-})) = d(0, R_1^{-})$.

Now we suppose that $\mathbb{K}$ has residue characteristic zero. Then, the function $f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}$ satisfies $\nu^+(f, \log r) = 1$ for every $r \in ]0, 1[$, hence $r > \frac{r^n}{n}$ for all $r \in ]0, 1[$, for every $n \in \mathbb{N}^*$. Therefore, $f$ has no zero different from 1 in $d(0, 1^{-})$ and, by Corollary B.14.10 of [7], is injective in $d(0, 1^{-})$. Then by Corollary B.13.10 of [7] we see that $\mathrm{Log}(d(1, 1^{-})) = d(0, 1^{-})$. This ends the proof. $\square$

**Corollary 1.A:** $\mathbb{K}$ is supposed to have residue characteristic 0. There is no root of 1 in $d(1, 1^{-})$, except 1.

*Proof.* Indeed any root of 1 should be a zero of $\mathrm{Log}$ in $d(1, 1^{-})$. $\square$

<figure><img src="image_2.png" /></figure>

**Notations:** If $\mathbb{K}$ has residue characteristic $p \neq 0$, we first denote by $\exp$ the inverse (or reciprocal) function of the restriction of $\mathrm{Log}$ to $d(1, R_1^{-})$, which obviously is a function defined in $d(0, R_1^{-})$, with values in $d(1, R_1^{-})$. If $\mathbb{K}$ has residue characteristic 0 we denote by $\exp$ the inverse function of $\mathrm{Log}$, which is obviously defined in $d(0, 1^{-})$ and takes values in $d(1, 1^{-})$.

**Theorem 1.3:** $\mathbb{K}$ is supposed to have residue characteristic $p \neq 0$ (resp. $p=0$). The function $\exp$ belongs to $\mathcal{A}_b(d(0, R_1^{-}))$ (resp. $\mathcal{A}_b(d(0, 1^{-}))$), is a bijection from $d(0, R_1^{-})$ onto $d(1, R_1^{-})$ (resp. from $d(0, 1^{-})$ onto $d(1, 1^{-}))$, and satisfies $\exp(x) = \exp'(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ whenever $x \in d(0, R_1^{-})$ (resp. $x \in d(0, 1^{-})$). Moreover, the disk of convergence of its series is equal to $d(0, R_1^{-})$ (resp. $d(0, 1^{-})$). Further, if $p \neq 0$, then $\exp$ is not an analytic element on $d(0, R_1^{-})$.

*Proof.* By Corollary B.14.15 of [7] we know that the function $\exp$ belongs to $\mathcal{A}_b(d(0, R_1^{-}))$ (resp. $\mathcal{A}_b(d(0, 1^{-}))$) and is obviously a bijection from $d(0, R_1^{-})$ onto $d(1, R_1^{-})$ (resp. from $d(0, 1^{-})$ onto $d(1, 1^{-}))$. As it is the reciprocal of $\mathrm{Log}$, it must satisfy $\exp(x) = \exp'(x)$ for all $x \in d(0, R_1^{-})$ (resp. $x \in d(0, 1^{-})$) and,
therefore, $\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ whenever $x \in d(0, R_1^-)$ (resp. $x \in d(0, 1^-)$). Thus the radius of convergence $r$ is at least $R_1$ (resp. 1). If the residue characteristic is 0, it is obviously seen that the series cannot converge for $|x| = 1$, hence the disk of convergence is $d(0, 1^-)$.

Now we suppose that the residue characteristic is $p \neq 0$. Suppose that the power series of $\exp$ converges in $d(0, R_1)$. Then $\exp$ has continuation to an analytic element element on $d(0, R_1)$. On the other hand, since $\nu(f, \log r) = 1$ for all $r \in ]0, R_1[$, we have $\nu^-(f, \log R_1) = 1$ and then by Theorem B.13.9 of [7] $\mathrm{Log}(d(1, R_1))$ is equal to $d(0, R_1)$. Hence, we can consider $\exp(\mathrm{Log}(x))$ in all the disk $d(0, R_1)$. By Corollary B.3.3 of [7] this is an analytic element element on $d(1, R_1)$. But this element is equal to the identity in all of $d(1, R_1^-)$ and, therefore, in all of $d(1, R_1)$. Of course this contradicts the fact that $\mathrm{Log}$ is not injective in the circle $C(1, R_1)$. This finishes proving that the disk of convergence of $\exp$ is just $d(0, R_1^-)$. $\square$

**Notations:** Henceforth, we put $e^x = \exp(x)$.

**Theorem 1.4:** $\mathbb{K}$ is supposed to have residue characteristic $p \neq 0$. Let $x \in d(0, R_1^-)$. Then $e^x$ is algebraic over $\mathbb{Q}_p$ if and only if so is $x$. Let $u \in d(0, 1^-)$. Then $\log(1+u)$ is algebraic over $\mathbb{Q}_p$ if and only if so is $u$.

*Proof.* By Theorem B.5.24 of [7], if $x$ is algebraic over $\mathbb{Q}_p$, so is $e^x$. Similarly, if $u$ is algebraic over $\mathbb{Q}_p$, so is $\log(1+u)$. Consequently, suppose that $e^x$ is algebraic over $\mathbb{Q}_p$. Then $e^x$ is of the form $1+t$ with $|t|<1$, hence $\log(1+t)$ is algebraic over $\mathbb{Q}_p$. But then, $\log(1+t) = \log(e^x) = x$, hence $x$ is algebraic over $\mathbb{Q}_p$. Now, more generally, suppose $\log(1+u)$ is algebraic over $\mathbb{Q}_p$, with $|u|<1$. Take $q \in \mathbb{N}$ such that $|p^q \log(1+u)| < R_1$. We have $p^q \log(1+u) = \log((1+u)^{p^q})$. Since $|p^q \log(1+u)| < R_1$, we have $|\log((1+u)^{p^q})| < R_1$, hence $\exp(\log((1+u)^{p^q})) = (1+u)^{p^q}$. Consequently, $(1+u)^{p^q}$ is algebraic over $\mathbb{Q}_p$ and hence so is $u$. $\square$

We can show a similar result when $p=0$.

**Theorem 1.5:** $\mathbb{K}$ is supposed to have residue characteristic 0. Let $x \in d(0, 1^-)$. Then $e^x$ is algebraic over $\mathbb{Q}_p$ if and only if so is $x$. Let $u \in d(0, 1^-)$. Then $\log(1+u)$ is algebraic over $\mathbb{Q}_p$ if and only if so is $u$.

The following proposition 1.6 will be used in the proof of Theorem 2.3 and is proven by induction, similarly as (1.4.2) in [16].

**Proposition 1.6:** Let $P_1, \dots, P_q \in \mathbb{K}[X]$ different from 0 and let $w_1, \dots, w_q \in \mathbb{K}$ be pairwise distinct. Let $F(x) = \sum_{j=1}^q P_j(x)e^{w_j x}$. Then $F$ is not identically zero.
## 2 Hermite-Lindemann's and Gel'fond-Schneider's Theorems in ultra-metric fields

We will use the following classical notation:

**Notation:** We will denote by $\mathcal{K}$ an algebraically closed complete ultrametric extension of $\mathbb{Q}$ of residue characteristic 0.

We will denote by $U$ the disk $d(0, 1)$ and by $D_0$ the disk $d(0, 1^{-})$ in the field $\mathbb{K}$ no matter what the residue characteristic.

If the residue characteristic of $\mathbb{K}$ is $p > 0$ we put $R_1 = p^{\frac{1}{p-1}}$ and denote by $D_1$ the disk $d(0, R_1^{-})$.

Given an algebraic number $a \in \mathbb{C}_p$ (resp. $a \in \mathcal{K}$) and $a_1, a_2, ..., a_q$ its conjugates over $\mathbb{Q}$ (with $a_1 = a$), we put $|a| = \max_{1 \le j \le q} |a_j|$ and we denote by $\text{den}(a)$ its smallest denominator, i.e. the smallest positive integer $q$ such that $qa$ is an algebraic integer. Then we put $s(a) = \max(\log|a|, \log(\text{den}(a)))$ and $s(a)$ is called the size of $a$. More generally we call denominator of a number $a$ all positive integer multiple of its smallest denominator.

Given a polynomial $P(X_1, ..., X_q) \in \mathbb{Z}[X_1, ..., X_q]$, we denote by $H(P)$ the supremum of the archimedean absolute values of its coefficients.

Given a positive real number $a$, we denote by $[a]$ the largest integer $n$ such that $n \le a$.

Hermite-Lindemann's theorem is well known in complex analysis. The same holds in p-adic analysis. The first proof was presented in 1930 by K. Malher [13]. This proof given in [13] is written in German and uses symbols which are not currently known. Here we present a new proof using classical methods in transcendental processes that are maybe easier to understand.

We will need Siegel's Lemma in all the following theorems of this chapter. We will choose a particular form of this famous lemma [16] whose formulation is due to M. Mignotte:

**Lemma 2.a (Siegel):** Let $E$ be a finite extension of $\mathbb{Q}$ of degree $q$ and let $\lambda_{i,j}$ $1 \le i \le m$, $1 \le j \le n$ be elements of $E$ integral over $\mathbb{Z}$. Let $M = \max(|\lambda_{i,j}|$ $1 \le i \le m$, $|1 \le j \le n|$) and let $(S)$ be the linear system $\{\sum_{j=1}^n \lambda_{i,j}x_j = 0, 1 \le i \le m\}$. There exists solutions $(x_1, ..., x_n)$ of $(S)$ such that $x_j \in \mathbb{Z}$ $\forall j = 1, ..., n$ and

$$ \log(|x_j|_{\infty}) \le \log(M) \frac{qm}{n-qm} + \frac{\log(2)}{2} \quad \forall j = 1, ..., n. $$

Lemma 2.b will be necessary in the proof of Theorem 2.4 and is easily proven in [16] since its proof implies no change in the field $\mathbb{K}$ since it only concerns algebraic numbers

**Lemma 2.b:** Let $a_1, ..., a_q \in \mathbb{K}$ be algebraic over $\mathbb{Q}$, let $P(X_1, ..., X_q) \in \mathbb{Z}[X_1, ..., X_q]$ be such that $\text{deg}_{X_j}(P) \le r_j$ $1 \le j \le q$ and let $\beta = P(a_1...a_q)$.
Then $\beta$ is algebraic over $\mathbb{Q}$, $d(a_1)^{r_1}...d(a_q)^{r_q}$ is a multiple of $\text{den}(\beta)$ and we have

$$s(\beta) \le \log H(P) + \sum_{j=1}^{q} (r_j s(a_j) + \log(r_j) + 1)$$

**Theorem 2.1 (Hermite-Lindemann):** Suppose that $\mathbb{K}$ has residue characteristic $p > 0$. Let $\alpha \in D_1$ be algebraic. Then $e^\alpha$ is transcendental.

*Proof.* We suppose that $\alpha$ and $e^\alpha$ are algebraic. Let $h = |\alpha|$. Let $E$ be the field $\mathbb{Q}[\alpha, e^\alpha]$, let $q = [E : \mathbb{Q}]$ and let $w$ be a common denominator of $\alpha$ and $e^\alpha$. We will construct a sequence of polynomials $(P_N(X, Y))_{N \in \mathbb{N}}$ in two variables such that $\text{deg}_X(P_N) = [\frac{N}{\log(N)}]$, $\text{deg}_Y(P_N) = [(\log N)^3]$ and such that the function $F_N(x) = P_N(x, e^x)$ satisfies further, for every $s = 0, ..., N-1$ and for every $j = 0, ..., \lfloor \log(N) \rfloor$

$$\frac{d^s}{dx^s} F_N(j\alpha) = 0.$$

According to formal computations in the proof of Hermite Lindemann's Theorem in the complex context, (Theorem 3.1.1 in [16]) we have

$$\frac{d^M F_N(\gamma_N)}{dx^M} = \sum_{l=0}^{u_1(N)} \sum_{m=0}^{u_2(N)} b_{l,m,N} \sum_{\sigma=0}^{u_1(N)} \left( \frac{u_1(N)!}{\sigma!(u_1(N)-\sigma)!} \right) \left( \frac{l!}{(u_1(N)-\sigma)!} \right) m^{u_1(N)-\sigma} \\ j^{u_1(N)-\sigma} \cdot (\alpha)^{u_1(N)-\sigma} \cdot (e^\alpha)^{j u_2(N)}.$$

We put $u_1(N) = \text{deg}_X(P_N)$, $u_2(N) = \text{deg}_Y(P_N)$. We will solve the system

$$w^{u_1(N)+u_2(N)} \frac{d^s}{dx^s} F_N(j\alpha) = 0, \quad 0 \le s \le N-1, \quad j = 0, ..., \lfloor \log(N) \rfloor$$

where the undeterminates are the coefficients $b_{l,m,N}$ of $P_N$. We then write the system under the form

$$\sum_{l=0}^{u_1(N)} \sum_{m=0}^{u_2(N)} b_{l,m,N} \sum_{\sigma=0}^{\min(s,l)} \left( \frac{s!}{\sigma!(s-\sigma)!} \right) \left( \frac{l!}{(l-\sigma)!} \right) m^{s-\sigma} \cdot j^{l-\sigma}.$$

$$(2) \qquad (w\alpha)^{l-\sigma} (we^\alpha)^{jm} \cdot w^{u_1(N)-(l-\sigma)+u_2(N)-jm} = 0.$$

That represents a system of $N[\log(N)]$ equations of at least $N([\log(N)])^2$ undeterminates, with coefficients in $E$, integral over $\mathbb{Z}$.

According to formal computations of Hermite-Lindemann's Theorem in the complex context (Theorem 3.1.1 in [16]), it appears that in the system (2), each

$$\text{factor } \left( \frac{s!}{\sigma!(s-\sigma)!} \right), \left( \frac{l!}{(l-\sigma)!} \right), m^{s-\sigma}, j^{l-\sigma}, (w\alpha)^{l-\sigma}, (we^\alpha)^{jm}, w^{u_1(N)-(l-\sigma)+u_2(N)-jm}$$
admits a bounding of the form $SN(\log(\log(N)))$ when $N$ goes to $+\infty$. On one hand $w^{u_1(N)+u_2(N)}$ is a common denominator and we have

$$ \log(w^{u_1(N)+u_2(N)}) \leq \log(\omega)\left(\frac{N}{\log(N)} + (\log(N))^3\right) $$

and hence we have a constant $T > 0$ such that

$$ (3) \qquad \log(w^{u_1(N)+u_2(N)}) \leq \frac{TM}{\log M}. $$

Next we notice that

$$ (4) \quad \log \left( \frac{u_1(N)!}{\sigma!(u_1(N)-\sigma)!} \right) \le u_1(N) \log(u_1(N)) \le \frac{N}{\log(N)} \log \left( \frac{N}{\log(N)} \right) \le N $$

and similarly,

$$ (5) \qquad \log \left( \frac{l!}{(u_1(N) - \sigma)!} \right) \le u_1(N) \log(u_1(N)) \le N. $$

and

$$ (6) \qquad \log(m^{u_1(N)-\sigma}) \le \frac{3N}{\log(N)} \log(\log(N)). $$

Now, we check that

$$ \log \left( j^{u_1(N)-\sigma} \cdot (|\bar{\alpha}|)^{u_1(N)-\sigma} \cdot (|\bar{e}^{\bar{\alpha}}|)^{j u_2(N)} \right) \le N + \frac{N}{\log(N)} \log(|\bar{\alpha}|) + \log(N) (\log(N))^3 \log(|\bar{e}^{\bar{\alpha}}|) $$

and hence there exists a constant $L > 0$ such that

$$ (7) \qquad \log \left( j^{u_1(N)-\sigma} \cdot (|\bar{\alpha}|)^{u_1(N)-\sigma} \cdot (|\bar{e}^{\bar{\alpha}}|)^{j u_2(N)} \right) \le LN. $$

Therefore by (2), (3), (4), (5), (6) and (7) we have a constant $C > 0$ such that each coefficient $a$ of the system satisfies

$$ (8) \qquad s(a) \le CN(\log(\log(N))). $$

By Siegel's Lemma 2.a and by (8) there exist integers $b_{l,m,N}$, $0 \le l \le u_1(N)$, $0 \le m \le u_2(N)$ in $\mathbb{Z}$ such that

$$ (9) \qquad 0 < \max_{l \le u_1(N), m \le u_2(N)} \log(|b_{l,m,N}|_\infty) \le \frac{qN \log(N)}{N(\log(N))^2 - qN \log(N)} (CN \log(\log(N))) $$

and such that the function

$$ (10) \qquad F_N(x) = \sum_{l=0}^{u_1(N)} \sum_{m=0}^{u_2(N)} b_{l,m,N} x^l e^{mx} $$
satisfies

$$ \frac{d^s}{dx^s} F_N(j\alpha) = 0, \ 0 \le s \le N-1, \ j = 0, 1, \dots, [\log(N)]. $$

Now, by (9), we can check that there exists a constant $G > 0$ such that

$$ (11) \qquad \max_{l \le u_1(N), m \le u_2(N)} \left( \log(|b_{l,m,N}|_\infty) \right) \le \frac{GN \log(\log(N))}{\log(N)}. $$

The function $F_N$ defined in (10) belongs to $\mathcal{A}(D_1)$ and is not identically zero, hence at least one of the numbers $\frac{d^s}{dx^s}F_N(0)$ is not null. Let $M$ be the biggest of the integers such that $\frac{d^s}{dx^s}F_N(j\alpha) = 0 \ \forall s = 0, \dots, M-1$, $j = 0, 1, 2, \dots, [\log(N)]$. Thus we have $M \ge N$ and there exists $j_0 \in \{0, 1, \dots, [\log(N)]\}$ such that $\frac{d^M}{dx^M}F_N(j_0\alpha) \ne 0$. We put $\gamma_N = \frac{d^M}{dx^M}F_N(j_0\alpha)$.

Let us now give an upper bound of $s(\gamma_N)$. On one hand $w^{u_1(N)+u_2(N)}$ is a common denominator and by (2) we have a constant $T > 0$ such that

$$ \log(w^{u_1(N)+u_2(N)}) \le \frac{TM}{\log M}. $$

On the other hand, by (1) we have

$$ \frac{d^M F_N(\gamma_N)}{dx^M} = \sum_{l=0}^{u_1(N)} \sum_{m=0}^{u_2(N)} b_{l,m,N} \sum_{\sigma=0}^{u_1(N)} \left( \frac{u_1(N)!}{\sigma!(u_1(N)-\sigma)!} \right) \left( \frac{l!}{(u_1(N)-\sigma)!} \right) m^{u_1(N)-\sigma}. $$ $$ j^{u_1(N)-\sigma} \cdot (\alpha)^{u_1(N)-\sigma} \cdot (e^{\alpha})^{ju_2(N)}. $$

Now, by (2), (3), (6), (7), (8), (10) and taking into account that the number of terms is bounded by $N(\log N)^2$, we can check that there exists a constant $B$ such that

$$ (12) \qquad s(\gamma_N) \le BN. $$

Let us now give an upper bound of $|\gamma_N|$. For convenience, we first suppose that $j_0 = 0$, hence $\frac{d^M}{dx^M}F_N(0) \ne 0$. Set $h = |\alpha|$. Then by Theorem B.9.1 of [7] we have $|\gamma_N| \le \frac{|F_N|(h)}{h^M}$. Moreover, we notice that $F_N$ admits at least $M[\log(M)]$ zeros in $d(0, h)$ and therefore by Corollary B.13.30 of [7] we have $|F_N|(h) \le \left(\frac{h}{R_1}\right)^{M[\log(M)]}$ because $|F_N|(r) \le 1 \ \forall r < R_1$. Consequently, $|\gamma_N| \le \frac{h^{M(\log(M)-1)}}{(R_1)^{M \log M}}$ and hence

$$ \log(|\gamma_N|) \le M(\log(M) - 1)(\log(h)) - M \log(M)(\log(R_1)). $$
Let $\lambda = \log(h) - \log(R_1)$. Then $\lambda < 0$. And we have $\log(|\gamma_N|) \le \lambda M \log(M) - M \log(h)$, therefore there exists a constant $A > 0$ such that

$$ (13) \qquad \log(|\gamma_N|) \le -AM \log(M). $$

Let us now stop assuming that $j_0 = 0$. Putting $z = x - j\alpha$ and $g(z) = f(x)$, since all points $j\alpha$ belong to $d(0, h)$, it is immediate to go back to the case $j_0 = 0$, which confirms (13) in the general case. But now, by Lemma A.8.10 in [7], relations (12) and (13) make a contradiction to the relation $-2qs(\gamma_N) \le \log(|\gamma_N|)$ satisfied by algebraic numbers and show that $\gamma_N$ is transcendental. But then, so is $e^\alpha$.

□

**Example:** Let $Q(x) \in \mathbb{Z}[x]$. Then $e^{pQ(p)}$ is transcendental. Moreover, if $Q$ is monic, and if $\alpha$ is a zero of $Q$, then $|p\alpha| \le \frac{1}{p}$ because $Q$ is monic and obviously $p\alpha$ is algebraic, hence $e^{p\alpha}$ is transcendental.

In the field of characteristic 0, $\mathcal{K}$ such as Levi-Civita's field [15], we have a similar version:

**Theorem 2.2:** Let $\alpha \in \mathcal{K}$ be algebraic, such that $|\alpha| < 1$. Then $e^\alpha$ is transcendental over $\mathbb{Q}$.

*Proof.* Everything works in $\mathcal{K}$ as in a field of residue characteristic $p \ne 0$ up to Relation (8) in the proof of Theorem 2.1. Here we can replace $R_1$ by 1 and therefore the conclusion is the same as in Theorem 2.1. □

Similarly as Hermite-Lindermann's Theorem, Gelfond-Schneider's Theorem is well known in the field $\mathbb{C}$ and has an analogue in an ultrametric field.

In the proof of Theorem 2.4 we will need the following theorem:

**Theorem 2.3:** Let $b_1, ..., b_n \in D_1$ (resp. in $D_0$). The functions $x, e^{b_1 x}, ..., e^{b_n x}$ are algebraically independent over $\mathbb{K}$ (resp. over $\mathcal{K}$) if and only if $b_1, ..., b_n$ are $\mathbb{Q}$-linearly independent.

**Theorem 2.4 (Gel'fond-Schneider):** $\mathbb{K}$ is supposed to have residue characteristic $p \ne 0$. Let $\ell \in D_1$, $\ell \ne 0$, and let $b \notin \mathbb{Q}$ belong to $\mathbb{K}$ be such that $b\ell \in D_1$. Then at least one of the three numbers $a = e^\ell$, $b$, $e^{b\ell}$ is transcendental.

*Proof.* A large part of the proof does not involve the topology of the field $\mathbb{K}$ and hence is similar to the proof in the field $\mathbb{C}$ [16] where we can copy many technical relations. We suppose that $a = e^\ell$, $b$ and $e^{b\ell}$ are algebraic over $\mathbb{Q}$. Let $L = \mathbb{Q}[e^\ell, b, e^{b\ell}]$ and let $\delta = [L : \mathbb{Q}]$ and let $d$ be a common denominator of $b, e^\ell, e^{b\ell}$.
Put $S = \max(1, |b|)$, $T \in S$, $\frac{R_1}{|\ell|} [\sigma = \log(\frac{T}{S})$, $\tau = \log T$, $\Lambda = d(0, S)$ and $\Delta = d(0, T)$. We will consider integers $N$ of the form $q^2$, with $q \in \mathbb{N}$ and we will first show that there exists a non-identically zero polynomial $P_N(X, Y) \in \mathbb{Z}[X, Y]$ such that $\deg_X(P_N) \le N^{\frac{3}{2}}$, and $\deg_Y(P_N) \le 2\delta N^{\frac{1}{2}}$ such that the function $F_N(x)$ defined in $\Delta$ by $F_N(x) = P_N(x, e^{\ell x})$ satisfy

$$F_N(i + jb) = 0 \quad \forall i = 1, \dots, N, \quad \forall j = 1, \dots, N.$$

In order to find $P_N$, let us write it

$$\sum_{h=0}^{N^{\frac{3}{2}}-1} \sum_{k=0}^{2\delta N^{\frac{1}{2}}-1} C_{h,k}(N) X^h Y^k$$

with $C_{h,k}(N) \in \mathbb{Z}$ and consider the system of equations where the $C_{h,k}(N)$ are the undeterminates:

$$d^{(4\delta+1)N^{\frac{3}{2}}}.F_N(i+jb) = 0 \quad (1 \le i \le N; 1 \le j \le N).$$

Thus, we obtain a system of $N^2$ equations of $2\delta N^2$ undeterminates in $\mathbb{Z}$, with coefficients in $L$. By Lemma 2.b, these coefficients have size bounded by

$$N^{\frac{3}{2}} \log(N) + N^{\frac{3}{2}} (8\delta + 2) \log(d) + \log(1 + |b|) + 2\delta \log(|e^{\ell+b\ell}|) \le \frac{3}{2} N^{\frac{3}{2}} \log(N).$$

By Lemma 2.a we can find in $\mathbb{Z}$ a family of integers not all equal to zero, $(C_{h,k}(N), 0 \le N^{\frac{3}{2}} - 1, 0 \le k \le 2\delta N^{\frac{1}{2}} - 1)$ satisfying

$$\log \left( \max_{h,k} |C_{h,k}(N)|_{\infty} \right) \le 2N^{\frac{3}{2}} \log N \left( \frac{\delta N^2}{2\delta N^2 - \delta N^2} \right) = 2N^{\frac{3}{2}} \log N$$

such that the function $F_N$ defined by $F_N(x) = P_N(x, e^{\ell x})$ satisfies $F_N(i+jb) = 0 \quad \forall i = 1, \dots, N, \quad \forall j = 1, \dots, N$.

Now we can check the function $F_N$ is an analytic element in every disk of the form $d(0, r)$ such that $r|\ell| < R_1$ and hence in $\Delta = d(0, T)$ [7]. Since the power of $x$ in the various terms is at most $N^{\frac{3}{2}}$ and since all coefficients are integers, we can check that $\log(|F_N|(T)) \le \tau N^{\frac{3}{2}}$. On the other hand, since the polynomial $P_N$ is not identically zero, by Proposition 1.6 $F_N$ is not identically zero and then, by classical results [7], the function $F_N$ has finitely many zeros in $\Lambda$. Particularly, there exists a point of the form $i+jb$ such that $F_N(i+jb) \ne 0$. Consequently there exists $M \ge N$ such that $F_N(i+jb) = 0 \quad \forall i \le M, \forall j \le M$ and there exists a point $\gamma_N$ of the form $i_0 + j_0b$ such that $F_N(\gamma_N) \ne 0$ with $M < i_0 \le M + 1, M < j_0 \le M + 1$. Consequently the number of zeros of $F_N$ in $\Lambda$ is at least $M^2$. Then by Corollary B.13.30 in [7] we have $\log(|F_N(\gamma_N)|) \le \tau N^{\frac{3}{2}} - \sigma M^2$, hence there exists $\lambda > 0$ such that

$$\log(|F_N(\gamma_N)|) \le -\lambda M^2 \quad \forall N \in \mathbb{N}.$$
By definition neither $\sigma$ nor $\tau$ depend on $N$, hence neither does $\lambda$.

On the other hand, by Lemma 2.b we can check that $s(F_N(\gamma_N))$ satisfies an inequality of the form $s(F_N(\gamma_N)) \le AM^{\frac{3}{2}} \log(M)$ which by (1) contradicts the inequality $-2\delta s(F_N(\gamma_N)) \le \log(|F_N(\gamma_N)|)$ and this ends the proof. $\square$

**Example:** Let $u = pe^p$ and let $\ell = pe^u$. Let $b \notin \mathbb{Q}$ be such that $|b| \le 1$. Then at least one of the 3 numbers $e^\ell$, $b$, $e^{b\ell}$ is transcendental. In particular, if $b$ is algebraic, then one of the numbers $e^\ell$ and $e^{b\ell}$ is transcendental.

**Theorem 2.5 (Gel'fond-Schneider in zero residue characteristic):** Let $K$ be an algebraically closed complete ultrametric field whose residue characteristic is 0. Let $\ell \in D_0$, $\ell \neq 0$, and let $b \notin \mathbb{Q}$ belong to $K$ and be such that $b\ell \in D_0$. Then at least one of the three numbers $a = e^\ell$, $b$, $e^{b\ell}$ is transcendental.

*Proof.* The proof is identical to the proof of Theorem 2.4 except that $T$ now belongs to $S, \frac{1}{|\ell|}[$.

$\square$

### 3 Nevanlinna Theory in $K$ and in an open disk

**Notations:** We denote by $M(K)$ the field of meromorphic functions in $K$ i.e. the field of fractions of $A(K)$. Let $d(a, R^{-})$ be a disk in $K$. We denote by $M(d(a, R^{-}))$ the field of fractions $A(d(a, R^{-}))$ and by $M_b(d(a, R^{-}))$ the field of fractions $A_b(d(a, R^{-}))$. Finally we put $M_u(d(a, R^{-})) = M(d(a, R^{-})) \setminus M_b(d(a, R^{-}))$.

Given two meromorphic functions $f, g \in M(K)$ or $f, g \in M(d(a, R^{-}))$ ($a \in K, R > 0$), we will denote by $W(f, g)$ the Wronskian of $f$ and $g$: $f'g - fg'$.

Let $f \in M(K) \setminus K(x)$ (resp. Let $f \in M_u(d(\alpha, R^{-}))$). A value $b \in K$ will be called a quasi-exceptional value for $f$ if $f - b$ has finitely many zeros in $K$ (resp. in $(\alpha, R^{-}))$) and it will be called an exceptional value for $f$ if $f - b$ has no zero in $K$ (resp. in $d(\alpha, R^{-}))$).

We have the following result:

**Theorem 3.1:** Let $f \in M(K)$ (resp. $f \in M_u(d(a, R^{-}))$). Then $f$ amits at most one quasi-exceptional value. Moreover, if $f \in A(K)$ (resp. $f \in A_u(d(a, R^{-}))$) then $f$ amits no quasi-exceptional value

The Nevanlinna Theory was made by Rolf Nevanlinna on complex functions [14], and widely used by many specialists of complex functions, particularly Walter Hayman [10]. 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 $f - b, b \in \mathbb{C}$.
A similar theory for functions in a p-adic field was constructed and correctly proved by A. Boutabaa [5] in the field $K$, after some previous work by Ha Huy Khoai [9]. See also [11]. In [6] the theory was extended to functions in $M(d(0, R^{-}))$ by taking into account Lazard's problem [12]. A new extension to functions out of a hole was made in [7] but we won't describe it because we would miss place. Here we will only give an abstract of the ultrametric Nevanlinna Theory in order to give the new theorems on $q$ small functions.

**Notations:** Recall that given three functions $\phi, \psi, \zeta$ defined in an interval $J =]a, +\infty[$ (resp. $J =]a, R[$), with values in $[0, +\infty[$, we shall write $\phi(r) \le \psi(r) + O(\zeta(r))$ if there exists a constant $b \in \mathbb{R}$ such that $\phi(r) \le \psi(r) + b\zeta(r)$. We shall write $\phi(r) = \psi(r) + O(\zeta(r))$ if $|\psi(r) - \phi(r)|$ is bounded by a function of the form $b\zeta(r)$.

Similarly, we shall write $\phi(r) \le \psi(r) + o(\zeta(r))$ if there exists a function $h$ from $J =]a, +\infty[$ (resp. from $J =]a, R[$) to $\mathbb{R}$ such that $\lim_{r \to +\infty} \frac{h(r)}{\zeta(r)} = 0$ (resp. $\lim_{r \to R} \frac{h(r)}{\zeta(r)} = 0$) and such that $\phi(r) \le \psi(r) + h(r)$. And we shall write $\phi(r) = \psi(r) + o(\zeta(r))$ if there exists a function $h$ from $J =]a, +\infty[$ (resp. from $J =]a, R[$) to $\mathbb{R}$ such that $\lim_{r \to +\infty} \frac{h(r)}{\zeta(r)} = 0$ (resp. $\lim_{r \to R} \frac{h(r)}{\zeta(r)} = 0$) and such that $\phi(r) = \psi(r) + h(r)$.

Throughout the next paragraphs, we will denote by $I$ the interval $[t, +\infty[$ and by $J$ an interval of the form $[t, R[$ with $t > 0$.

We have to introduce the counting function of zeros and poles of $f$, counting or not multiplicity. Here we will choose a presentation that avoids assuming that all functions we consider admit no zero and no pole at the origin.

**Definitions:** Next, let $f = \frac{h}{l} \in \mathcal{M}(\mathbb{K})$ (resp. $f = \frac{h}{l} \in \mathcal{M}(d(a, R^{-})))$. The order of a zero $\alpha$ of $f$ will be denoted by $\omega_{\alpha}(f)$. Next, given any point $\alpha \in \mathbb{K}$ resp. $\alpha \in d(a, R^{-}))$, the number $\omega_{\alpha}(h) - \omega_{\alpha}(l)$ does not depend on the functions $h, l$ choosed to make $f = \frac{h}{l}$. Thus, we can generalize the notation by setting $\omega_{\alpha}(f) = \omega_{\alpha}(h) - \omega_{\alpha}(l)$. We then denote by $Z(r, f)$ the counting function of zeros of $f$ in $d(0, r)$ in the following way.

Let $(a_n)$, $1 \le n \le \sigma(r)$ be the finite sequence of zeros of $f$ such that $0 < |a_n| \le r$, of respective order $s_n$.

We set $Z(r, f) = \max(\omega_0(f), 0) \log r + \sum_{n=1}^{\sigma(r)} s_n (\log r - \log |a_n|)$ and so, $Z(r, f)$ is called the *counting function of zeros of $f$ in $d(0, r)$, counting multiplicity*.

In order to define the counting function of zeros of $f$ without multiplicity, we put $\overline{\omega}_0(f) = 0$ if $\omega_0(f) \le 0$ and $\overline{\omega}_0(f) = 1$ if $\omega_0(f) \ge 1$.

Now, we denote by $\bar{Z}(r, f)$ the counting function of zeros of $f$ without multiplicity:
$\bar{Z}(r, f) = \bar{\omega}_0(f) \log r + \sum_{n=1}^{\sigma(r)} (\log r - \log |a_n|)$ and so, $\bar{Z}(r, f)$ is called the counting function of zeros of $f$ in $d(0, r)$ ignoring multiplicity.

In the same way, considering the finite sequence $(b_n)$, $1 \le n \le \tau(r)$ of poles of $f$ such that $0 < |b_n| \le r$, with respective multiplicity order $t_n$, we put

$N(r, f) = \max(-\omega_0(f), 0) \log r + \sum_{n=1}^{\tau(r)} t_n(\log r - \log |b_n|)$ and then $N(r, f)$ is called the counting function of the poles of $f$, counting multiplicity.

Next, in order to define the counting function of poles of $f$ without multiplicity, we put $\bar{\omega}_0(f) = 0$ if $\omega_0(f) \ge 0$ and $\bar{\omega}_0(f) = 1$ if $\omega_0(f) \le -1$ and we set

$\bar{N}(r, f) = \bar{\omega}_0(f) \log r + \sum_{n=1}^{\tau(r)} (\log r - \log |b_n|)$ and then $\bar{N}(r, f)$ is called the counting function of the poles of $f$, ignoring multiplicity.

Now we can define the Nevanlinna function $T(r, f)$ in $I$ or $J$ as $T(r, f) = \max(Z(r, f), N(r, f))$ and the function $T(r, f)$ is called characteristic function of $f$ or Nevanlinna function of $f$.

Finally, if $S$ is a subset of $K$ we will denote by $Z_0^S(r, f')$ the counting function of zeros of $f'$, excluding those which are zeros of $f - a$ for any $a \in S$.

**Remark:** If we change the origin, the functions $Z, N, T$ are not changed, up to an additive constant.

In a $p$-adic field such as $K$, the first Main Theorem is almost immediate.

**Theorem 3.2:** Let $f \in \mathcal{M}(K)$ (resp. $f \in \mathcal{M}(d(0, R^{-}))$) have no zero and no pole at 0. Then $\log(|f|(r)) = \log(|f(0)|) + Z(r, f) - N(r, f)$.

Then we can derive Theorem 3.3 (Theorem C.4.3 in [7])

**Theorem 3.3:** Let $f, g \in \mathcal{M}(K)$ (resp. $f, g \in \mathcal{M}(d(0, R^{-}))$). Then $Z(r, fg) \le Z(r, f) + Z(r, g)$, $N(r, fg) \le N(r, f) + N(r, g)$, $T(r, fg) \le T(r, f) + T(r, g)$, $T(r, f+g) \le T(r, f) + T(r, g) + O(1)$, $T(r, cf) = T(r, f) \forall c \in K^*$, $T(r, \frac{1}{f}) = T(r, f)$, $T(r, \frac{f}{g}) \le T(r, f) + T(r, g)$.

Suppose now $f, g \in \mathcal{A}(K)$ (resp. $f, g \in \mathcal{A}(d(0, R^{-}))$). Then $Z(r, fg) = Z(r, f) + Z(r, g)$, $T(r, f) = Z(r, f)$, $T(r, fg) = T(r, f) + T(r, g) + O(1)$ and $T(r, f+g) \le \max(T(r, f), T(r, g))$. Moreover, if $\lim_{r \to +\infty} T(r, f) - T(r, g) = +\infty$ then $T(r, f+g) = T(r, f)$ when $r$ is big enough.

**Corollary 3.A:** Let $f \in \mathcal{M}(K)$ (resp. $f \in \mathcal{M}(d(0, R^{-}))$). Then

$$Z(r, \frac{f'}{f}) - N(r, \frac{f'}{f}) \le -\log r + O(1).$$
Thus we have Theorem 3.4 (Theorem C.4.8 in [7])

**Theorem 3.4 (First Main Fundamental Theorem):** Let $f, g \in \mathcal{M}(\mathbb{K})$ (resp. let $f, g \in \mathcal{M}(d(0, R^{-}))$). Then $T(r, f+b) = T(r, f) + O(1)$. Let $h$ be a Moebius function. Then $T(r, f) = T(r, h \circ f) + O(1)$. Let $P(X) \in \mathbb{K}[X]$. Then $T(r, P(f)) = \deg(P)T(r, f) + O(1)$ and $T(r, f'P(f) \ge T(r, P(f))$.

Suppose now $f, g \in \mathcal{A}(\mathbb{K})$ (resp. $f, g \in \mathcal{A}(d(0, R^{-}))$). Then $Z(r, fg) = Z(r, f) + Z(r, g)$, $T(r, f) = Z(r, f)$, $T(r, fg) = T(r, f) + T(r, g) + O(1)$ and $T(r, f+g) \le \max(T(r, f), T(r, g))$. Moreover, if $\lim_{r \to +\infty} T(r, f) - T(r, g) = +\infty$ then $T(r, f+g) = T(r, f)$ when $r$ is big enough.

The following Theorem 3.5 is a good way to obtain the famous Second Main Theorem (Theorem C.4.24 in [7]).

**Theorem 3.5:** Let $f \in \mathcal{M}(\mathbb{K})$ and let $a_1, \dots, a_q \in \mathbb{K}$ be distinct. Then

$$ (q-1)T(r, f) \le \max_{1 \le k \le q} \left( \sum_{j=1, j \ne k}^{q} Z(r, f - a_j) \right) + O(1). $$

**Theorem 3.6 (Second Main Theorem, Theorem C.4.24 in [7]):** Let $\alpha_1, \dots, \alpha_q \in \mathbb{K}$, with $q \ge 2$, let $S = \{\alpha_1, \dots, \alpha_q\}$ and let $f \in \mathcal{M}(\mathbb{K})$ (resp. $f \in \mathcal{M}_u(d(0, R^{-}))$). Then

$$ (q-1)T(r, f) \le \sum_{j=1}^{q} \bar{Z}(r, f - \alpha_j) + \bar{N}(r, f) - Z_0^S(r, f') - \log r + O(1) \quad \forall r \in I \quad (\text{resp. } \forall r \in J). $$

Now we can easily deduce the following corollaries:

**Corollary 3.B:** Let $a_1, a_2 \in \mathbb{K}$ ($a_1 \ne a_2$) and let $f, g \in \mathcal{A}(\mathbb{K})$ satisfy $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$ ($i=1, 2$). Then $f = g$.

**Remark:** Corollary 3.B does not hold in complex analysis. Indeed, let $f(z) = e^z$, $g(z) = e^{-z}$, let $a_1 = 1$, $a_2 = -1$. Then $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$ ($i=1, 2$), though $f \ne g$.

**Corollary 3.C:** Let $a_1, a_2, a_3 \in \mathbb{K}$ ($a_i \ne a_j \forall i \ne j$) and let $f, g \in \mathcal{A}_u(d(a, R^{-}))$ (resp. $f, g \in \mathcal{A}_u(D)$) satisfy $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$ ($i=1, 2, 3$). Then $f = g$.

**Corollary 3.D:** Let $a_1, a_2, a_3, a_4 \in \mathbb{K}$ ($a_i \ne a_j \forall i \ne j$) and let $f, g \in \mathcal{M}(\mathbb{K})$ satisfy $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$ ($i=1, 2, 3, 4$). Then $f = g$.

**Corollary 3.E:** Let $a_1, a_2, a_3, a_4, a_5 \in \mathbb{K}$ ($a_i \ne a_j \forall i \ne j$) and let $f, g \in \mathcal{M}_u(d(a, R^{-}))$ (resp. $f, g \in \mathcal{M}_u(D)$) satisfy $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$ ($i=1, 2, 3, 4, 5$). Then $f = g$.
**Remark:** Let $f(x) = \frac{x}{3x-1}$, $g(x) = \frac{x^2}{x^2+2x-1}$. Let $a_0 = 0$, $a_1 = 1$, $a_2 = \frac{1}{2}$. Then we can check that $f^{-1}(\{a_i\}) = g^{-1}(\{a_i\})$, $i = 1, 2, 3$. So, Corollary 3.D is sharp.

# 4 Exceptional values of meromorphic functions and derivatives

The paragraph is aimed at studying various properties of derivatives of meromorphic functions, particularly their sets of zeros [2], [3], [4]. Many important results are due to Jean-Paul Bézivin [1], [2].

We will first notice a general property concerning quasi-exceptional values of meromorphic functions and derivatives.

**Theorem 4.1:** Let $f \in \mathcal{M}(\mathbb{K}) \setminus \mathbb{K}(x)$ (resp. Let $f \in \mathcal{M}_u(d(\alpha, R^{-}))$). If $f$ admits a quasi-exceptional value, then $f'$ has no quasi-exceptional value different from 0.

*Proof.* Without loss of generality, we may assume $\alpha = 0$ and that $f$ has no zero and no pole at 0. Let $b \in \mathbb{K}$ and suppose that $b$ is a quasi-exceptional value of $f$. There exist $P \in \mathbb{K}[x]$ and $l \in \mathcal{A}(\mathbb{K}) \setminus \mathbb{K}[x]$ (resp. and $l \in \mathcal{A}_u(d(0, R^{-}))$) without common zeros, such that $f = b + \frac{P}{l}$.

Let $c \in \mathbb{K}^*$. Remark that $f' - c = \frac{P'l - Pl' - cl^2}{l^2}$. Let $a \in \mathbb{K}$ (resp. let $a \in d(0, R^{-})$). If $a$ is a pole of $f$, it is a pole of $f' - c$ and we can check that
$$ (1) \quad \omega_a(P'l - Pl' - cl^2) = \omega_a(l') = \omega_a(l) - 1 $$
because $a$ is not a zero of $P$.

Now suppose that $a$ is not a pole of $f$. Then

$$ (2) \quad \omega_a(f' - c) = \omega_a(P'l - Pl' - cl^2) $$

Consequently, $Z(r, f' - c) = Z(r, (P'l - Pl' - cl^2) | l(x) \neq 0)$. But, by (1) we have

$$ (3) \quad Z(r, (P'l - Pl' - cl^2) | l(x) = 0) < Z(r, l). $$

and therefore by (2) and (3) we obtain

$$ (4) \quad Z(r, f' - c) = Z(r, (P'l - Pl' - cl^2) | l(x) \neq 0) > Z(r, P'l - Pl' - cl^2) - Z(r, l) $$

Now, let us examine $Z(r, P'l - Pl' - cl^2)$. Let $r \in ]0, +\infty[$ (resp. let $r \in ]0, R[$). Since $l \in \mathcal{A}(\mathbb{K})$ is transcendental (resp. since $l \in \mathcal{A}_u(d(0, R^{-}))$, we can check that when $r$ is big enough, we have $|Pl'|(r) < |c|(l|(r))^2$ and $|Pl|(r) < |c|(l|(r))^2$, hence clearly $|P'l - Pl'|(r) < |c|(l|(r))^2$ and hence $|P'l - Pl' - cl^2|(r) = |c|(l|(r))^2$. Consequently, when $r$ is big enough, by Theorem C.4.2 in [7] we have $Z(r, P'l - Pl' - cl^2) = Z(r, l^2) + O(1)$. But $Z(r, l^2) = 2Z(r, l)$, hence $Z(r, P'l - Pl' - cl^2) = 2Z(r, l) + O(1)$ and therefore by (4) we check that when $r$ is big enough, we obtain
(5) $Z(r, f' - c) > Z(r, l)$.

Now, if $l \in \mathcal{A}(\mathbb{K})$, since $l$ is transcendental, by (5), for every $q \in \mathbb{N}$, we have $Z(r, f' - c) > Z(r, l) > q \log r$, when $r$ is big enough, hence $f' - c$ has infinitely many zeros in $\mathbb{K}$. And similarly if $l \in \mathcal{A}_u(d(0, R^{-}))$, then by (5), $Z(r, f' - c)$ is unbounded when $r$ tends to $R$, hence $f' - c$ has infinitely many zeros in $d(0, R^{-})$. $\square$

We will now notice a property of differential equations of the form $y^{(n)} - \psi y = 0$ that is almost classical.

The problem of a constant Wronskian is involved in several questions.

**Theorem 4.2:** Let $h$, $l \in \mathcal{A}(\mathbb{K})$ (resp. $h$, $l \in \mathcal{A}(d(\alpha, R^{-}))$) and satisfy $h'l - hl' = c \in \mathbb{K}$, with $h$ non-affine. If $h$, $l$ belong to $\mathcal{A}(\mathbb{K})$, then $c = 0$ and $\frac{h}{l}$ is a constant. If $c \neq 0$ and if $h$, $l \in \mathcal{A}(d(\alpha, R^{-}))$, there exists $\phi \in \mathcal{A}(d(\alpha, R^{-}))$ such that $h'' = \phi h$, $l'' = \phi l$.

*Proof.* Suppose $c \neq 0$. If $h(a) = 0$, then $l(a) \neq 0$. Next, $h$ and $l$ satisfy
(1) $\frac{h''}{h} = \frac{l''}{l}$.

Remark first that since $h$ is not affine, $h''$ is not identically zero. Next, every zero of $h$ or $l$ of order $\ge 2$ is a trivial zero of $h'l - hl'$, which contradicts $c \ne 0$. So we can assume that all zeros of $h$ and $l$ are of order 1.

Now suppose that a zero $a$ of $h$ is not a zero of $h''$. Since $a$ is a zero of $h$ of order 1, $\frac{h''}{h}$ has a pole of order 1 at $a$ and so does $\frac{l''}{l}$, hence $l(a) = 0$, a contradiction. Consequently, each zero of $h$ is a zero of order 1 of $h$ and is a zero of $h''$ and hence, $\frac{h''}{h}$ is an element $\phi$ of $M(\mathbb{K})$ (resp. of $M(d(\alpha, R^{-})))$ that has no pole in $\mathbb{K}$ (resp. in $d(\alpha, R^{-}))$. Therefore $\phi$ lies in $\mathcal{A}(\mathbb{K})$ (resp. in $\mathcal{A}(d(\alpha, R^{-})))$.

The same holds for $l$ and so, $l''$ is of the form $\psi l$ with $\psi \in \mathcal{A}(\mathbb{K})$ (resp. in $\mathcal{A}(d(\alpha, R^{-})))$. But since $\frac{h''}{h} = \frac{l''}{l}$, we have $\phi = \psi$.

Now, suppose $h$, $l$ belong to $\mathcal{A}(\mathbb{K})$. Since $h''$ is of the form $\phi h$ with $\phi \in \mathcal{A}(\mathbb{K})$, we have $|h''|(r) = |\phi|(r)|h|(r)$. But by Theorem C.2.10 in [7], we know that $|h''|(r) \le \frac{1}{r^2}|h|(r)$, a contradiction when $r$ tends to $+\infty$. Consequently, $c = 0$. But then $h'l - hl' = 0$ implies that the derivative of $\frac{h}{l}$ is identically zero, hence $\frac{h}{l}$ is constant. $\square$

**Corollary 4.A :** Let $h$, $l \in \mathcal{A}(\mathbb{K})$ with coefficients in $\mathbb{Q}$, also be entire functions in $\mathbb{C}$, with $h$ non-affine. If $h'l - hl'$ is a constant $c$, then $c = 0$.
**Theorem 4.3:** Let $\psi \in \mathcal{M}(\mathbb{K})$ (resp. let $\psi \in \mathcal{M}_u(d(\alpha, R^{-}))$) and let $(\mathcal{E})$ be the differential equations $y'' - \psi y = 0$. Let $E$ be the sub-vector space of $\mathcal{A}(\mathbb{K})$ (resp. of $\mathcal{A}(d(\alpha, R^{-}))$) of the solutions of $(\mathcal{E})$. Then, the dimension of $E$ is 0 or 1.

*Proof.* Suppose $E$ is not $\{0\}$. Let $h, l \in E$ be non-identically zero. Then $h''l - hl'' = 0$ and therefore $h'l - hl'$ is a constant $c$. On the other hand, since $h, l$ are not identically zero, neither are $h''$, $l''$. Therefore, $h, l$ are not affine functions.

Suppose $\psi$ belongs to $\mathcal{M}(\mathbb{K})$ and that $h, l$ belong to $\mathcal{A}(\mathbb{K})$. By Theorem 4.2, we have $c = 0$ and hence $\frac{h}{l}$ is a constant, which proves that $E$ is of dimension 1.

Suppose now that $\psi$ lies in $\mathcal{M}_u(d(\alpha, R^{-}))$ and that $h, l$ belong to $\mathcal{A}(d(\alpha, R^{-}))$. If $\psi$ lies in $\mathcal{A}(d(\alpha, R^{-}))$, then by Theorem 4.1, $E = \{0\}$. Finally, suppose that $\psi$ lies in $\mathcal{M}_u(d(\alpha, R^{-})) \setminus \mathcal{A}(d(\alpha, R^{-}))$. If $c \neq 0$, by Theorem 4.2, there exists $\phi \in \mathcal{A}(d(\alpha, R^{-}))$ such that $h'' = \phi h, l'' = \phi l$. Consequently, $\phi = \psi$, hence $\psi \in \mathcal{A}(\mathbb{K})$ and therefore $c = 0$. Hence $h'l - hl' = 0$ again and hence $\frac{h}{l}$ is a constant. Thus, we see that $E$ is at most of dimension 1. $\square$

**Remark:** The hypothesis $\psi$ unbounded in $d(\alpha, R^{-})$ is indispensable to show that the space $E$ is of dimension 0 or 1, as shows the example given again by the p-adic hyperbolic functions $h(x) = \cosh(x)$ and $l(x) = \sinh(x)$. The radius of convergence of both $h, l$ is $p^{\frac{1}{p-1}}$ when $\mathbb{K}$ has residue characteristic $p$ and is 1 when $\mathbb{K}$ has residue characteristic 0. Of course, both functions are solutions of $y'' - y = 0$ but they are bounded.

The following Theorem 4.4 is an improvement of Theorem 4.2. It follows previous results [1].

**Theorem 4.4 [2]:** Let $f, g \in \mathcal{A}(\mathbb{K})$ be such that $W(f,g)$ is a non-identically zero polynomial. Then both $f, g$ are polynomials.

*Proof.* First, by Theorem 4.2 we check that the claim is satisfied when $W(f,g)$ is a polynomial of degree 0. Now, suppose the claim holds when $W(f,g)$ is a polynomial of certain degree $n$. We will show it for $n+1$. Let $f,g \in \mathcal{A}(\mathbb{K})$ be such that $W(f,g)$ is a non-identically zero polynomial $P$ of degree $n+1$.

Thus, by hypothesis, we have $f'g - fg' = P$, hence $f''g - fg'' = P'$. We can extract $g'$ and get $g' = \frac{f'g-P}{f}$. Now consider the function $Q = f''g' - f'g''$ and replace $g'$ by what we just found: we can get $Q = f'\left(\frac{f''g-fg''}{f}\right) - \frac{Pf''}{f}$.

Now, we can replace $f''g - fg''$ by $P'$ and obtain $Q = \frac{f'P' - Pf''}{f}$. Thus, in that expression of $Q$, we can write $|Q|(R) \le \frac{|f|(R)|P|(R)}{R^2|f|(R)}$, hence $|Q|(R) \le$
$\frac{|P|(R)}{R^2} \forall R > 0$. But by definition, $Q$ belongs to $\mathcal{A}(\mathbb{K})$. Consequently, $Q$ is a polynomial of degree $t \le n - 1$.

Now, suppose $Q$ is not identically zero. Since $Q = W(f', g')$ and since $\deg(Q) < n$, by the induction hypothesis $f'$ and $g'$ are polynomials and so are $f, g$. Finally, suppose $Q = 0$. Then $P'f' - Pf'' = 0$ and therefore $f', P$ are two solutions of the differential equation of order 1 for meromorphic functions in $\mathbb{K}: (\mathcal{E})$ $y' = \psi y$ with $\psi = \frac{P'}{P}$, whereas $y$ belongs to $\mathcal{A}(\mathbb{K})$. By Theorem 4.3, the space of solutions of $(\mathcal{E})$ is of dimension 0 or 1. Consequently, there exists $\lambda \in \mathbb{K}$ such that $f' = \lambda P$, hence $f$ is a polynomial. The same holds for $g$. $\square$

Here we can find again the following result that is known and may be proved without ultrametric properties:

Let $F$ be an algebraically closed field and let $P, Q \in F[x]$ be such that $PQ' - P'Q$ is a constant $c$, with $\deg(P) \ge 2$. Then $c = 0$.

**Notation:** Let $f \in \mathcal{A}(\mathbb{K})$. We can factorize $f$ in the form $\bar{f}\tilde{f}$ where the zeros of $\bar{f}$ are the distinct zeros of $f$ each with order 1. Moreover, if $f(0) \ne 0$ we will take $\bar{f}(0) = 1$.

**Lemma 4.a:** Let $U, V \in \mathcal{A}(\mathbb{K})$ have no common zero and let $f = \frac{U}{V}$. If $f'$ has finitely many zeros, there exists a polynomial $P \in \mathbb{K}[x]$ such that $U'V - UV' = P\tilde{V}$

*Proof.* If $V$ is a constant, the statement is obvious. So, we assume that $V$ is not a constant. Now $\tilde{V}$ divides $V'$ and hence $V'$ factorizes in the way $V' = \tilde{V}Y$ with $Y \in \mathcal{A}(\mathbb{K})$. Then no zero of $Y$ can be a zero of $V$. Consequently, we have

$$f'(x) = \frac{U'V - UV'}{V^2} = \frac{U'\bar{V} - UY}{\bar{V}^2\tilde{V}}.$$

The two functions $U'\bar{V} - UY$ and $\bar{V}^2\tilde{V}$ have no common zero since neither have $U$ and $V$. So, the zeros of $f'$ are those of $U'\bar{V} - UY$ which therefore has finitely many zeros and consequently is a polynomial. $\square$

**Theorem 4.5:** Let $f \in \mathcal{M}(\mathbb{K})$ have finitely many multiple poles, such that for certain $b \in \mathbb{K}$, $f' - b$ has finitely many zeros. Then $f$ belongs to $\mathbb{K}(x)$.

*Proof.* Suppose first $b = 0$. Let us write $f = \frac{U}{V}$ with $U, V \in \mathcal{A}(\mathbb{K})$, having no common zeros. By Lemma 4.a, there exists a polynomial $P \in \mathbb{K}[x]$ such that $U'V - UV' = P\tilde{V}$. Since $f$ has finitely many multiple poles, $\tilde{V}$ is a polynomial, hence so is $U'V - UV'$. But then by Theorem 4.4, both $U, V$ are polynomials, which ends the proof when $b = 0$. Consider now the general case. $f' - b$ is the derivative of $f - bx$ that satisfies the same hypothesis, so the conclusion is immediate. $\square$
**Notation:** For each $n \in \mathbb{N}^*$, we set $\lambda_n = \max\{\frac{1}{|k|}, 1 \le k \le n\}$. Given positive integers $n, q$, we denote by $C_n^q$ the combination $\frac{n!}{q!(n-q)!}$. Let us recall that $\log$ is the Neperian logarithm, we denote by $e$ the number such that $\log(e) = 1$ and Exp is the real exponential function.

**Remark:** For every $n \in \mathbb{N}^*$, we have $\lambda_n \le n$ because $k|k| \ge 1 \forall k \in \mathbb{N}$. The equality holds for all $n$ of the form $p^h$.

Lemmas 4.b and 4.c are due to Jean-Paul Bézivin [1]:

**Lemma 4.b:** Let $U, V \in \mathcal{A}(d(0, R^{-}))$. Then for all $r \in ]0, R[$ and $n \ge 1$ we have

$$|U^{(n)}V - UV^{(n)}|(r) \le |n!|\lambda_n \frac{|U'V - UV'|(r)}{r^{n-1}}.$$

More generally, given $j, l \in \mathbb{N}$, we have

$$|U^{(j)}V^{(l)} - U^{(l)}V^{(j)}|(r) \le |(j!)(l!)|\lambda_{j+l} \frac{|U'V - UV'|(r)}{r^{j+l-1}}$$

**Lemma 4.c:** Let $U, V \in \mathcal{A}(\mathbb{K})$ and let $r, R \in ]0, +\infty[$ satisfy $r < R$. For all $x, y \in \mathbb{K}$ with $|x| \le R$ and $|y| \le r$, we have the inequality:

$$|U(x + y)V(x) - U(x)V(x + y)| \le \frac{R|U'V - UV'|(R)}{e(\log R - \log r)}.$$

**Notation:** Let $f \in \mathcal{M}(d(0, R^{-}))$. For each $r \in ]0, R[$, we denote by $\zeta(r, f)$ the number of zeros of $f$ in $d(0, r)$, taking multiplicity into account and set $\xi(r, f) = \zeta(r, \frac{1}{f})$. Similarly, we denote by $\beta(r, f)$ the number of multiple zeros of $f$ in $d(0, r)$, each counted with its multiplicity and we set $\gamma(r, f) = \beta(r, \frac{1}{f})$.

**Theorem 4.6** [2] Let $f \in \mathcal{M}(\mathbb{K})$ be such that for some $c, q \in ]0, +\infty[$, $\gamma(r, f)$ satisfies $\gamma(r, f) \le cr^q$ in $[1, +\infty[$. If $f'$ has finitely many zeros, then $f \in \mathbb{K}(x)$.

*Proof.* Suppose $f'$ has finitely many zeros and set $f = \frac{U}{V}$. If $V$ is a constant, the statement is immediate. So, we suppose $V$ is not a constant and hence it admits at least one zero $a$. By Lemma 4.a, there exists a polynomial $P \in \mathbb{K}[x]$ such that $U'V - UV' = P\tilde{V}$. Next, we take $r, R \in [1, +\infty[$ such that $|a| < r < R$ and $x \in d(0, R)$, $y \in d(0, r)$. By Lemma 4.c we have

$$|U(x + y)V(x) - U(x)V(x + y)| \le \frac{R|U'V - UV'|(R)}{e(\log R - \log r)}.$$

Notice that $U(a) \ne 0$ because $U$ and $V$ have no common zero. Now set $l = \max(1, |a|)$ and take $r \ge l$. Setting $c_1 = \frac{1}{e|U(a)|}$, we have

$$|V(a + y)| \le c_1 \frac{R|P(R)|\tilde{V}|(R)}{\log R - \log r}.$$
**Corollary 4.C:** Let $f \in \mathcal{M}(\mathbb{K}) \setminus \mathbb{K}(x)$ be such that $\xi(r, f) \le cr^q$ in $[1, +\infty[$ for some $c, q \in ]0, +\infty[$. Then for each $k \in \mathbb{N}^*$, $f^{(k)}$ has no quasi-exceptional value.

*Proof.* Indeed, if $k=1$, the statement just comes from Corollary 4.B Now suppose $k \ge 2$. Each pole $a$ of order $n$ of $f$ is a pole of order $n+k$ of $f^{(k)}$ and $f^{(k)}$ has no other pole. Consequently, we have $\gamma(r, f^{k-1}) = \xi(r, f^{(k-1)}) \le kcr^q$. So, we can apply Corollary 4.B to $f^{(k-1)}$ to show the claim. $\square$

Theorem 4.6 suggests us the following conjecture:

**Conjecture:** Let $f \in \mathcal{M}(\mathbb{K})$ be such that $f'$ admits finitely many zeros. Then $f \in \mathbb{K}(x)$.

In other words, the conjecture suggests that the derivative of a meromorphic function in $\mathbb{K}$ has no quasi-exceptional value, except if it is a rational function.

**Remark:** Of course, there exist meromorphic functions in $\mathbb{K}$ having no zero but not satisfying the hypotheses of Theorem 4.6, hence such a function cannot have primitives. For example, consider an entire function $f$ having an infinity of zeros $(a_n)_{n \in \mathbb{N}}$ of order 2 such that $|a_n| < |a_{n+1}|$, $\lim_{n \to +\infty} |a_n| = +\infty$ and $2n \le |a_n|$. Then the meromorphic function $g = \frac{1}{f}$ has no zeros but does not satisfy the hypotheses of Theorem 4.6 hence it has no primitives.

# 5 Small functions

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 3 small functions. Here we will recall the construction of a similar theory.

**Definitions and notation:** Throughout the chapter we set $a \in \mathbb{K}$ and $R \in ]0, +\infty[$. For each $f \in \mathcal{M}(\mathbb{K})$ (resp. $f \in \mathcal{M}(d(a, R^{-}))$) we denote by $\mathcal{M}_f(\mathbb{K})$, (resp. $\mathcal{M}_f(d(a, R^{-})))$ the set of functions $h \in \mathcal{M}(\mathbb{K})$, (resp. $h \in \mathcal{M}(d(a, R^{-})))$ such that $T(r, h) = o(T(r, f))$ when $r$ tends to $+\infty$ (resp. when $r$ tends to $R$). Similarly, if $f \in \mathcal{A}(\mathbb{K})$ (resp. $f \in \mathcal{A}(d(a, R^{-})))$ we shall denote by $\mathcal{A}_f(\mathbb{K})$ (resp. $\mathcal{A}_f(d(a, R^{-})))$ the set $\mathcal{M}_f(\mathbb{K}) \cap \mathcal{A}(\mathbb{K})$, (resp. $\mathcal{M}_f(d(a, R^{-})) \cap \mathcal{A}(d(a, R^{-})))$.

The elements of $\mathcal{M}_f(\mathbb{K})$ (resp. $\mathcal{M}_f(d(a, R^{-})))$ are called small meromorphic functions with respect to $f$, (small functions in brief). Similarly, if $f \in \mathcal{A}(\mathbb{K})$ (resp. $f \in \mathcal{A}(d(a, R^{-})))$ the elements of $\mathcal{A}_f(\mathbb{K})$ (resp. $\mathcal{A}_f(d(a, R^{-})))$ are called small analytic functions with respect to $f$, (small functions in brief).

Theorems 5.1 and Theorem 5.2 are immediate consequences of Theorems C.9.1 and C.9.2 in [7]:
**Theorem 5.1:** Let $a \in \mathbb{K}$ and $r > 0$. Then $\mathcal{A}_f(\mathbb{K})$ is a $\mathbb{K}$-subalgebra of $\mathcal{A}(\mathbb{K})$, $\mathcal{A}_f(d(a, R^-))$ is a $\mathbb{K}$-subalgebra of $\mathcal{A}(d(a, R^-))$ $\mathcal{M}_f(\mathbb{K})$ is a subfield field of $\mathcal{M}(\mathbb{K})$, $\mathcal{M}_f(d(a, R^-))$ is a subfield of field of $\mathcal{M}(a, R^-)$. Moreover, $\mathcal{A}_b(d(a, R^-)$ is a sub-algebra of $\mathcal{A}_f(d(a, R^-)$ and $\mathcal{M}_b(d(a, R^-)$ is a subfield of $\mathcal{M}_f(d(a, R^-)$.

**Theorem 5.2:** Let $f \in \mathcal{M}(\mathbb{K})$, $(\text{resp. } f \in \mathcal{M}(d(0, R^-)))$ and let $g \in \mathcal{M}_f(\mathbb{K})$, $(\text{resp. } g \in \mathcal{M}_f(d(0, R^-)))$. Then $T(r, fg) = T(r, f) + o(T(r, f))$ and $T(r, \frac{f}{g}) = T(r, f) + o(T(r, f))$, $(\text{resp. } T(r, fg) = T(r, f) + o(T(r, f))$ and $T(r, \frac{f}{g}) = T(r, f) + o(T(r, f)))$.

Theorem 5.3 is known as Second Main Theorem on Three Small Functions in p-adic analysis [7] and [10]. It holds as well as in complex analysis, where it was showed first and it is proven in the same way.

**Theorem 5.3:** Let $f \in \mathcal{M}(\mathbb{K})$ $(\text{resp. } f \in \mathcal{M}_u(d(0, R^-)))$ and let $w_1, w_2, w_3 \in \mathcal{M}_f(\mathbb{K})$ $(\text{resp. } w_1, w_2, w_3 \in \mathcal{M}_f(d(0, R^-)))$ be pairwise distinct. Then $T(r, f) \le \sum_{j=1}^3 \bar{Z}(r, f - w_j) + o(T(r, f))$, resp $T(r, f) \le \sum_{j=1}^3 \bar{Z}(r, f - w_j) + o(T(r, f))$, resp. $T_R(r, f) \le \sum_{j=1}^3 \bar{Z}_R(r, f - w_j) + o(T(r, f))$.

**Theorem 5.4:** Let $f \in \mathcal{M}(\mathbb{K})$ $(\text{resp. } f \in \mathcal{M}_u(d(0, R^-)))$ and let $w_1, w_2 \in \mathcal{M}_f(\mathbb{K})$ $(\text{resp. } w_1, w_2 \in \mathcal{M}_f(d(0, R^-)))$ be distinct. Then $T(r, f) \le \bar{Z}(r, f - w_1) + \bar{Z}(r, f - w_2) + o(T(r, f))$, $(\text{resp. } T(r, f) \le \bar{Z}(r, f - w_1) + \bar{Z}(r, f - w_2) + o(T(r, f)))$.

*Proof.* Suppose first $f \in \mathcal{M}(\mathbb{K})$ or $f \in \mathcal{M}_u(d(0, R^-))$. Let $g = \frac{1}{f}$, $h_j = \frac{1}{w_j}$, $j = 1, 2, h_3 = 0$. Clearly,

$$T(r, g) = T(r, f) + O(1), \quad T(r, h) = T(r, w_j), \quad j = 1, 2,$$

so we can apply Theorem 5.3 to $g, h_1, h_2, h_3$. Thus we have: $T(r, g) \le \bar{Z}(r, g - h_1) + \bar{Z}(r, g - h_2) + \bar{Z}(r, g) + o(T(r, g))$.

But we notice that $\bar{Z}(r, g - h_j) = \bar{Z}(r, f - w_j)$ for $j = 1, 2$ and $\bar{Z}(r, g) = \bar{N}(r, f)$. Moreover, we know that $o(T(r, g)) = o(T(r, f))$. Consequently, the claim is proved when $w_1w_2$ is not identically zero.

Now, suppose that $w_1 = 0$. Let $\lambda \in \mathbb{K}^*$, let $l = f + \lambda$ and $\tau_j = u_j + \lambda$, $(j = 1, 2, 3)$. Thus, we have $T(r, l) = T(r, f) + O(1)$, $T(r, \tau_j) = T(r, w_j) + O(1)$, $(j = 1, 2)$, $\bar{N}(r, l) = \bar{N}(r, f)$. By the claim already proven whenever $w_1w_2 \neq 0$ we may write $T(r, l) \le \bar{Z}(r, l - \tau_1) + \bar{Z}(r, l - \tau_2) + \bar{N}(r, l) + o(T(r, l)))$ hence $T(r, f) \le \bar{Z}(r, f - w_1) + \bar{Z}(r, f - w_2) + \bar{N}(r, l) + o(T(r, f)))$.

□

Next, by setting $g = f - w_1$ and $w = w_1 + w_2$, we can write Corollary 5.A:
**Corollary 5.A:** Let $g \in \mathcal{M}(\mathbb{K})$ (resp. $g \in \mathcal{M}_u(d(0, R^{-}))$) and let $w \in \mathcal{M}_g(\mathbb{K})$. Then $T(r, g) \le \bar{Z}(r, g) + \bar{Z}(r, g - w) + \bar{N}(r, g) + o(T(r, g))$, (resp. $T(r, g) \le \bar{Z}(r, g) + \bar{Z}(r, g - w) + \bar{N}(r, g) + o(T(r, g))$).

**Corollary 5.B:** Let $f \in \mathcal{A}(\mathbb{K})$ (resp. $f \in \mathcal{A}_u(d(0, R^{-}))$) and let $w_1, w_2 \in \mathcal{A}_f(\mathbb{K})$ (resp. $w_1, w_2 \in \mathcal{A}_f(d(0, R^{-}))$) be distinct. Then $T(r, f) \le \bar{Z}(r, f - w_1) + \bar{Z}(r, f - w_2) + o(T(r, f))$ ($r \to +\infty$), resp. ($r \to R^{-}$).

And similarly to Corollary 5.A, we can get Corollary 5.C:

**Corollary 5.C:** Let $f \in \mathcal{A}(\mathbb{K})$ (resp. $f \in \mathcal{A}_u(d(0, R^{-}))$, resp. $f \in \mathcal{A}^c(D)$) and let $w \in \mathcal{A}_f(\mathbb{K})$. Then $T(r, f) \le \bar{Z}(r, f) + \bar{Z}(r, f - w) + o(T(r, f))$, (resp. $T(r, f) \le \bar{Z}(r, f) + \bar{Z}(r, f - w) + o(T(r, f))$).

We are now able to state a theorem on $q$ small functions that is not as good as Yamanoi's Theorem [17] in complex analysis, but seems the best possible in ultrametric analysis;

**Theorem 5.5 [8] (A. Escassut, C.C. Yang):** Let $f \in \mathcal{M}(\mathbb{K})$ be transcendental (resp. $f \in \mathcal{M}_u(d(0, R^{-}))$) and let $w_j \in \mathcal{M}_f(\mathbb{K})$ ($j = 1, ..., q$) (resp. $w_j \in \mathcal{M}_f(d(a, R^{-}))$) be $q$ distinct small functions other than the constant $\infty$. Then

$$qT(r, f) \le 3 \sum_{j=1}^{q} \bar{Z}(r, f - w_j) + o(T(r, f)),$$

(resp.

$$qT(r, f) \le 3 \sum_{j=1}^{q} \bar{Z}(r, f - w_j) + o(T(r, f)),$$

Moreover, if $f$ has finitely many poles in $\mathbb{K}$ (resp. in $d(0, R^{-})$), then

$$qT(r, f) \le 2 \sum_{j=1}^{q} \bar{Z}(r, f - w_j) + o(T(r, f)),$$

(resp.

$$qT(r, f) \le 2 \sum_{j=1}^{q} \bar{Z}(r, f - w_j) + o(T(r, f)).$$

*Proof.* By Theorem 5.3, for every triplet $(i, j, k)$ such that $1 \le i \le j \le k \le q$, we can write

$$T(r, f) \le \bar{Z}(r, f - w_i) + \bar{Z}(r, f - w_j) + \bar{Z}(r, f - w_k) + o(T(r, f)).$$

The number of such inequalities is $C_q^3$. Summing up, we obtain
$$ (1) \qquad C_q^3 T(r, f) \le \sum_{(i,j,k), 1 \le i \le j \le k \le q} \bar{Z}(r, f-w_i) + \bar{Z}(r, f-w_j) + \bar{Z}(r, f-w_k) + o(T(r, f)). $$

In this sum, for each index $i$, the number of terms $\bar{Z}(r, f - w_i)$ is clearly $C_{q-1}^2$. Consequently, by (1) we obtain

$$ C_q^3 T(r, f) \le C_{q-1}^2 \sum_{i=1}^q \bar{Z}(r, f - w_i) + o(T(r, f)) $$

and hence

$$ \frac{q}{3} T(r, f) \le \sum_{i=1}^{q} \bar{Z}(r, f - w_i) + o(T(r, f)). $$

Suppose now that $f$ has finitely many poles. By Theorem 5.4, for every pair $(i, j)$ such that $1 \le i \le j \le q$, we have

$$ T(r, f) \le \bar{Z}(r, f - w_i) + \bar{Z}(r, f - w_j) + o(T(r, f)). $$

The number of such inequalities is then $C_q^2$. Summing up we now obtain

$$ (2) \qquad C_q^2 T(r, f) \le \sum_{(i,j, 1 \le i \le j \le q)} \bar{Z}(r, f - w_i) + \bar{Z}(r, f - w_j) + o(T(r, f)). $$

In this sum, for each index $i$, the number of terms $\bar{Z}(r, f - w_i)$ is clearly $C_{q-1}^1 = q - 1$. Consequently, by (1) we obtain

$$ C_q^2 T(r, f) \le (q-1) \sum_{i=1}^q \bar{Z}(r, f - w_i) + o(T(r, f)) $$

and hence

$$ \frac{q}{2} T(r, f) \le \sum_{i=1}^{q} \bar{Z}(r, f - w_i) + o(T(r, f)). $$

□

**Definition:** Let $f, g \in \mathcal{M}(\mathbb{K})$ (resp. $f, g \in \mathcal{M}_u(d(a, R^-))$). Then $f$ and $g$ will be to share a small function, I.M. $w \in \mathcal{M}(\mathbb{K})$ (resp. $w \in \mathcal{M}(d(a, R^-))$) if $f(x) = w(x)$ implies $g(x) = w(x)$ and if $g(x) = w(x)$ implies $f(x) = w(x)$.

**Theorem 5.6:** Let $f, g \in \mathcal{M}(\mathbb{K})$ be transcendental (resp. $f, g \in \mathcal{M}_u(d(a, R^-))$) be distinct and share $q$ distinct small functions I.M. $w_j \in \mathcal{M}_f(\mathbb{K}) \cap \mathcal{M}_g(\mathbb{K})$ ($j = 1, ..., q$) (resp. $w_j \in \mathcal{M}_f(d(a, R^-)) \cap \mathcal{M}_g(d(a, R^-))$ ($j = 1, ..., q$)) other than the constant $\infty$. Then

$$ \sum_{j=1}^{q} \bar{Z}(r, f - w_j) \le \bar{Z}(r, f - g) + o(T(r, f)) + o(T(r, g)). $$
*Proof.* Suppose that $f$ and $g$ belong to $\mathcal{M}(\mathbb{K})$, are distinct and share $q$ distinct small functions I.M. $w_j \in \mathcal{M}_f(\mathbb{K}) \cap \mathcal{M}_g(\mathbb{K})$ ($j=1, ..., q$).

Lat $b$ be a zero of $f - w_i$ for a certain index $i$. Then it is also a zero of $g - w_i$. Suppose that $b$ is counted several times in the sum $\sum_{j=1}^q \bar{Z}(r, f - w_j)$, which means that it is a zero of another function $f - w_h$ for a certain index $h \neq i$. Then we have $w_i(b) = w_h(b)$ and hence $b$ is a zero of the function $w_i - w_h$ which belongs to $\mathcal{M}_f(\mathbb{K})$. Now, put $\tilde{Z}(r, f - w_1) = \bar{Z}(r, f - w_1)$ and for each $j > 1$, let $\tilde{Z}(r, f - w_j)$ be the counting function of zeros of $f - w_j$ in the disk $d(0, r^{-})$ ignoring multiplicity and avoiding the zeros already counted as zeros of $f - w_h$ for some $h < j$. Consider now the sum $\sum_{j=1}^q \tilde{Z}(r, f - w_j)$. Since the functions $w_i - w_j$ belong to $\mathcal{M}_f(\mathbb{K})$, clearly, we have

$$ \sum_{j=1}^{q} \bar{Z}(r, f - w_j) = \sum_{j=1}^{q} \tilde{Z}(r, f w_j) = o(T(r, f)) $$

It is clear, from the assumption, that $f(x) - w_j(x) = 0$ implies $g(x) - w_j(x) = 0$ and hence $f(x) - g(x) = 0$. Since $f - g$ is not the identically zero function, it follows that

$$ \sum_{j=1}^{q} \bar{Z}(r, f - w_j) \le \bar{Z}(r, f - g). $$

Consequently,

$$ \sum_{j=1}^{q} \bar{Z}(r, f - w_j) \le \bar{Z}(r, f - g) + o(T(r, f)) + o(T(r, g)). $$

Now, if $f$ and $g$ belong to $\mathcal{M}(d(0, R^{-}))$ the proof is exactly the same. $\square$

<figure><img src="image_3.png" /></figure>

**Theorem 5.7 [8] (A. Escassut, C.C. Yang):** Let $f, g \in \mathcal{M}(\mathbb{K})$ be transcendental (resp. $f, g \in \mathcal{M}_u(d(a, R^{-}))$) be distinct and share 7 distinct small functions (other than the constant $\infty$) I.M. $w_j \in \mathcal{M}_f(\mathbb{K}) \cap \mathcal{M}_g(\mathbb{K})$ ($j=1, ..., 7$) (resp. $w_j \in \mathcal{M}_f(d(a, R^{-})) \cap \mathcal{M}_g(d(a, R^{-}))$, resp. $w_j \in \mathcal{M}_f(D) \cap \mathcal{M}_g(D)$ ($j=1, ..., 7$), ). Then $f = g$.

Moreover, if $f$ and $g$ have finitely many poles and share 3 distinct small functions (other than the constant $\infty$) I.M. then $f = g$.

*Proof.* We put $M(r) = \max(T(r, f), T(r, g))$. Suppose that $f$ and $g$ are distinct and share $q$ small function I.M. $w_j$, ($1 \le j \le q$). By Theorem 5.5, we have

$$ qT(r, f) \le 3 \sum_{j=1}^{q} \bar{Z}(r, f - w_j) + o(T(r, f)). $$
But thanks to Theorem 5.6, we can derive

$$qT(r, f) \leq 3T(r, f - g) + o(T(r, f))$$

and similarly

$$qT(r, g) \leq 3T(r, f - g) + o(T(r, g))$$

hence

$$(1) \qquad qM(r) \leq 3T(r, f - g) + o(M(r)).$$

By Theorem C.4.8 in [7], we can derive that

$$qM(r) \leq 3(T(r, f) + T(r, g)) + o(M(r))$$

and hence $qM(r) \leq 6M(r) + o(M(r))$. That applies to the situation when $f$ and $g$ belong to $\mathcal{M}(\mathbb{K})$ as well as when when $f$ and $g$ belong to $\mathcal{M}_u(d(0, R^{-}))$. Consequently, it is impossible if $q \geq 7$ and hence the first statement of Theorem 5.7 is proved.

Suppose now that $f$ and $g$ have finitely many poles. By Theorems C.4.8 in [7], Relation (1) gives us

$$qM(r) \leq 2M(r) + o(M(r))$$

which is obviously absurd whenever $q \geq 3$ and proves that $f = g$ when $f$ and $g$ belong to $\mathcal{M}(\mathbb{K})$ as well as when $f$ and $g$ belong to $\mathcal{M}_u(d(0, R^{-}))$. $\square$

**Corollary 5.D:** Let $f, g \in \mathcal{A}(\mathbb{K})$ be transcendental (resp. $f, g \in \mathcal{A}_u(d(a, R^{-}))$) be distinct and share 3 distinct small functions (other than the constant $\infty$) I.M. $w_j \in \mathcal{A}_f(\mathbb{K}) \cap \mathcal{A}_g(\mathbb{K})$ ($j = 1, 2, 3$) (resp. $w_j \in \mathcal{A}_f(d(a, R^{-})) \cap \mathcal{A}_g(d(a, R^{-}))$, ($j = 1, 2, 3$)). Then $f = g$.

## References

[1] Bezivin, J.-P., Wronskien et equations differentielles p-adiques, Acta Arith., 158, no. 1, 6178 (2013).

[2] Bezivin, J.-P., Boussaf, K. and Escassut, A. A. Zeros of the derivative of a p-adic meromorphic function, Bull. Sci. Math., 136, no. 8, 839847 (2012).

[3] Boussaf, K. Picard values of p-adic meromorphic functions, p-Adic Numbers Ultrametric Anal. Appl., 2, no. 4, 285292 (2010).

[4] Boussaf, K., Ojeda, J. and Escassut, A. Primitives of p-adic meromorphic functions, Contemp. Math., 551, 5156 (2011).

[5] Boutabaa, A. Théorie de Nevanlinna p-adique, Manuscripta Math. 67, p. 251-269 (1990).
[6] Boutabaa, A. and Escassut, A. *URS and URSIMS for p-adic meromorphic functions inside a disk*, Proc. of the Edinburgh Mathematical Society 44, p. 485-504 (2001).

[7] Escassut, A. *p-adic Analytic Functions*. World Scientific Publishing Co. Pte. Ltd. Singapore, (2021).

[8] Escassut, A. and Yang, C.C. *A short note on two p-adic meromorphic functions sharing a few small ones*, Rendiconti del Circolo Matematico di Palermo 70 (2), p. 623-630.

[9] Ha H. K. *On p-adic meromorphic functions*, Duke Mathematical Journal, 50, 695-711 (1983).

[10] Hayman, W. K. *Meromorphic Functions*. Oxford University Press, (1975)

[11] Hu, P.C. and Yang, C.C. *Meromorphic Functions over non-Archimedean Fields*, Kluwer Academic Publishers, (2000).

[12] Lazard, M. *Les zéros des fonctions analytiques sur un corps valué complet*, IHES, Publications Mathématiques no. 14, pp. 47-75 (1962).

[13] Malher, K. *Ein Beweis der Transzendenz der P-adischen Exponentialfunktion*, J. reine angew. Math., 169, pp. 61-66 (1932).

[14] Nevanlinna, R. *Le théorème de Picard-Borel et la théorie des fonctions méromorphes*. Gauthiers-Villars, Paris (1929).

[15] Shamseddine, K. *A brief survey of the study of power series and analytic functions on the Levi-Civita fields*, Contemporary Mathematics Volume 596, p. 269-279, (2013).

[16] Waldschmidt, M. *Nombres transcendants*, Lecture Notes in Mathematics, 402, Springer Verlag (1974).

[17] Yamanoi, K. *The second main theorem for small functions and related problems* Acta Mathematica 192, p. 225-294 (2004).

Laboratoire de Mathématiques Blaise Pascal
UMR CNRS 6620
Université Clermont Auvergne
F 63000 Clermont-Ferrand
France