$G$-FUNCTIONS, $G$-OPERATORS, AND DIOPHANTINE APPROXIMATIONS (Diophantine phenomena in differential equations and dynamical systems)

$G$-FUNCTIONS, $G$-OPERATORS, AND

DIOPHANTINE

APPROXIMATIONS

永田誠 (MAKOTO NAGATA)

京都大学数理解析研究所 (Research

Institute

for

Mathematical

Sciences, Kyoto University)

ABSTRACT. The aim ofthis artcle istointroduceanoutline oftopics on$G$-functions. Fordetails, pleaserefer

totheREFERENCES

CONTENTS

50

the definitionof$G$-functions, examples

51

$G$-functions and Diophantineapproximations (Ql and Q2)

1-0 E-functions

1-1 $G$-functions, Galchkin’s result

1-2 two problems (Ql and Q2)

\S 2

$G$-operators (an

answer

ofQl)

2-1 notations and definitions

2-2 Chudnovsky’s result

2-3 $\mathrm{A}\mathrm{n}\mathrm{d}\mathrm{r}\mathrm{e}" \mathrm{s}$ result

2-4 ageneralization ofG-0perators

\S 3

$G$-functions and Diophantine approximations, revisited (an

answer

of Q2)

3-0 algebraicfunctions

31

G-functions

\S 4

further problems

\S 0

THE DEFINITION OF $G$_{-FUNCTIONS, EXAMPLES}

A $G$-function is a power series solution, with a certain arithmetical condition, of

a

linear differential

equation

over a

rationalfunction field.

Let$K$ be

a

number field

as

the basefield. (i.e., the degreeof$K$

over

$\mathbb{Q}$is finite. $[K$: $\mathbb{Q}]<\infty$.)

But for simplicity,

we

always

assume

that $K$ is the rational numberfield $\mathbb{Q}:K=$ Q.

Definition. (G-functions)

We

call$y=y(x)$

a

$G$-function if

(0) $y$satisfies

a

lineardifferentialequation $/K(x)$

82

and$y$ is representedby

a

power series

over

$K$;

$y= \sum_{i=0}^{\infty}\alpha_{i}x^{i}$ $\in K[[x]]$.

$($1$)^{1}$ there exists

a

positive constant, $C$ (which is independent of_$i=1,2$,

$\ldots$) such that

(G-1) $|\alpha_{\mathrm{i}}|\leq C’$ for$i=1,2$,

$\ldots$

For $i=0,1,2$,

.

.

.

’ let$d_{i}\in \mathrm{N}$ be the minimum

common

denominatorof$\alpha_{0}$,$\alpha_{1}$,

.

. .

,$\alpha_{i}$

.

(i.e., $d_{i}$ is the minimum positive integer suchthat $d_{i}\alpha_{0}$,$d_{i}\alpha_{1}$,

. ..

’$d_{i}\alpha_{i}\in \mathbb{Z}$)

Then

(G-2) $|d\mathrm{j}$ $\leq C^{i}$ for$i=1,$2,

$\ldots$

$\square$

This is

the

definition

ofG-functions.

By this definition, namely by this condition (G-2), in general, it is not

so

easy to determine whether

a

givenfunction is

a

$G$-function

or

not.

Now, we show

some

examples ofG-functions.

Examples ofG-functions.

(1) The logarithmic function, polylogarithms: $\log(1-x)=-\sum_{i=1}^{\infty}\underline{x}i$, _$Lk\{\%$) $:= \sum_{i=1}^{\infty x_{\mathrm{F}}}l\neg.\cdot$; _$k\in$N.

These

are

G-functions.

To verify the condition (G-2) in the definition,

we

use

“the prime number theorem.”

(2) Algebraic functions $\in K[[x]]$ defined

over

$\mathbb{Q}$ (arealso G-functions).

The second example (2)

comes

from s0-called Eisenstein’s theorem relatedto the radiiof

convergence.

For further information,

see

[BC].

(3) Gauss’s hypergeometirc series with rationalparameters (are

also

$G$ function ,

$2F1( \alpha, \beta, \gamma;x):=\sum_{=j0}^{\infty}\frac{(\alpha)_{i}(\beta)_{\dot{1}}}{(\gamma)_{i}i!}x^{i}$ where $\alpha,\mathrm{d}$,$\gamma\in \mathbb{Q}$

.

Here

$(\alpha)_{0}=1$

,

$(\alpha)_{i}=\alpha(\alpha+1)\cdots$$(\alpha+i-1)$ for $i\in$ N.

We

note that the parameters $\alpha,\beta,\gamma$

are

in $\mathbb{Q}$

,

not

$K$

.

Of course, $\mathrm{y}$ is assumed to notbe

a

negative integer

nor

zero.

The reason ofthethird example (3) is similar to thefirst example (1).

We note that, in general, it is known that the rationality ofthe parameters $\alpha$,$\beta$,$\gamma$ is not unnecessary.

Forfurtherinfomation,

see

[A], and for the rationalityof the parameters,

see

[Ba], [H].

Remark.

The

exponential function$e^{x}= \sum_{i=0}^{\infty}\frac{x}{i!}\dot{.}$ is not

a

G-function.

Proof

of

$” log(l -x)$

satisfies

(G-2)”.

Let$d_{n}$ be the lowest

common

multipleof 1, 2, 3,

. .

.

,_$n$. Then

$d_{n}:=p$$p \leq n\prod_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}},p^{e_{\mathrm{p}}}$

($e_{p}:=$

max{e

$|$

$\mathrm{B}$ $m\in\{1$,

$\ldots$ ,$n\}\mathrm{s}.\mathrm{t}$. $p^{e_{p}}|m$

}

$\}$ $\mathrm{S}.\prod_{p\leq n}p^{\log_{\mathrm{p}}n}=\prod_{p\leq n}n\sim n^{n/\log n}p.\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}$

Thelast relation

comes

from “the prime number theorem.”

Then

$\log d_{n}\sim\frac{n}{1\mathrm{o}\mathrm{g}n}\log n=n$ $\Rightarrow$ $d_{n}\sim e^{n}$

.

Therefore

the condition (G-2)

holds for

$\log(1-x)$

.

$\square$

\S 1

$G$_{-FUNCTIONS AND DIOPHANTINE APPROXIMATIONS} ($\mathrm{Q}$$1$ AND $\mathrm{Q}2$)

In this section, this word, Diophantine approxi mations,

means

Diophantine approximations about some

values of$G$-functions at

some

points.

We willintroduce

a

result, the original isas Galochkin’s result. This is similar to aresult ofE-functions,

which

are an

object for the transcendental number theory.

After introducing both results,

_we

propose two questions. Ql and Q2. The aim ofthis section is to set

down Ql and Q2. In the other sections,

we

willconsider these two questions.

1-0

G-FUNCTIONS

There is another class of solutions of

linear differential

equations, $E$-functions. The definition is similar to

that of$G$-functions. An $E$-funcitonis also

a

power series solution of(eq), with another certain aritmetical

condition.

In short, the classof$E$-functions, in fact, looks like

a

generalization of the exponential function $e^{x}$

.

This article is, of course, for $G$-functions, not for $E$-functions. So

we

will skip the definition of

E-$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}.2$ We

will only introduce

a

propertyof E-functions.

But anyway, the exponential function $e^{x}$ is atypical example of

E-functions.3

Fordetails, see [Sh].

Before showing

a

propertyof$E$-functions,

we

introduce

a

linear differential equation of

a matrix

type.

We considerthe linear differentialequationofamatrix type (instead of(eq))

(EQ) $\frac{d}{dx}m=Am$, _{$A\in M_{n}(K(x))$}, $m$:

a

vector solution

Ifevery componentof$m$satisfiestheconditions (G-1) and (G-2) in the definitionof$G$-functions,wealso

call it

a

$G$-function(,

or

_$m$is a$G$-function vector).

As well

as

$E$-functions, ifevery component of$m$ satisfies the arithmetical condition of the definition of

$E$-funcitons,we also call it

an

E-function.

Now the next theorem is aproperty ofE-functions.

Theorem E. (Cf. [Sh])

Assume

that

every

components

_of

$m=\mathrm{t}(f_{1}$,

. . .

,$f_{n})$ in (EQ)$(n\geq 2)$ is

an

$E$

-function.

And

assume

that

$f_{1}$,

$\ldots$ ,$f_{n}$

are

linearlyindependent

over

$\mathbb{C}(x)$

.

Then

for

anypositive real $i$$\epsilon>0,$ and

for

$\forall\xi\in \mathbb{Q}$ but

finite

exceptions, there eists

a

constant$\exists c_{1}$ such that

$|H\mathrm{v}_{1}$$( \xi)+\cdots+H_{n}f_{n}(\xi)|>\frac{H}{H^{n+\epsilon}}$ holds

for

$\forall H_{1}$,

$\ldots$ ,$H_{n}\in \mathbb{Z}$ with$H:= \max_{i}|H$;$|\geq c_{1}$

.

(Here $c_{1}$ depends

on

$A,$$m$,$\epsilon$,$\xi$.) $\square$

One may

come

up with that this is

an

analogous to Lindemann’stheorem in the transcendental number

theory.

(Lindemann’s theorem) Let $\alpha_{1}$,

.

.

.

,$\alpha_{n}\in K$ be distinct, then for $\forall H_{1}$,

$\ldots$ ,$H_{n}\in K:$ not all zero,

we

have

$H_{1}e^{\alpha_{1}}1-$

.

$..+H_{n}e",$ $\neq 0.$ $\square$

(Thatis to say, thenumbers $e^{\alpha_{1}}$,

.

..

’$e^{\alpha_{n}}$

are

linearly independent

over

$K.$)

We know thatLindemann’s theorem allows the

case

$K=$ _Q.

$2\mathrm{A}\mathrm{n}E$-function is a powerseries solution of (eq),

$y$$= \sum_{\dot{\mathrm{a}}=0}\infty-+\alpha_{l}$

. $x^{\dot{|}}\in K[[x]]$ with (1)forany$\epsilon>0$, $|0$

.

$|=O(:\epsilon|.)$ for$iarrow\infty$.

(2) let$d_{\dot{l}}$ bethecommondenominatorof$\alpha 0$,$\ldots$ ,$\alpha$., then$|\mathrm{d}_{\mathrm{i}}|=O(i^{\epsilon:})$ for$iarrow\infty$. Here$f(i)=$O(g(i))for$iarrow$oo means3$G$

$\mathrm{s}.\mathrm{t}$

.

$\mathrm{f}(\mathrm{i})\leq Cg(i)$ for $i=0,1,2$,$\ldots$.

84

1-1 G-FUNCTIONS

It

seems

that the original ofthe following Theorem is as aresult of Galochkin.

In order to clarify

our

problems,

we

modify it slightly. Butthe essential is notmodified. For details, see

[G] and also [C].

Theorem 1. ([G]. See also [C])

Let$m=\mathrm{t}$(Q1)

$\ldots$ ,$f_{n}$) be

a

vector solution in (EQ)$(n\geq 2)$

.

Assume that$f_{1}$,

$\ldots$ ,$f_{n}$

are

$G$-functions, and

assume

that$fi$,$\ldots$ ,$f_{n}$

are

linearly independent

over

$\mathbb{C}(x)$

.

If

(EQ) is

a

G-Operator,

(We willgive the

_definition

_of

$G$-operators inthe

next

section.)

for

any

positive real$\forall\epsilon>0,$ there exists constant$\exists c_{1}$ such that thefollowing holds:

for

any $q\in \mathbb{Z}$ with $|q|\geq c_{1}$,

there exists constant$\exists c_{2}$ such that

$|H_{1}f1(1fq)+\cdot$$..+H_{n}f_{n}(1/q)|> \frac{H}{H^{n+\epsilon}}$ $h\mathit{0}lds$

for

$\forall H_{1}$,

$\ldots$ ,$H_{n}\in \mathbb{Z}$ with$H:= \max_{i}|H_{i}|$ $\geq c_{2}$

.

(Here $c_{1}$ depends

on

$m$, $A$, $\epsilon$, and$c_{2}$ depends

on

$m$, $A$,

$\epsilon$, $q.)$ $\square$

One

might

see

thatthisTheorem 1 issimilar to TheoremE.

But

we

havetwo questions.

1-2 TWO PROBLEMS

Problems.

(Q1) What is

a

$G- operator^{Q}$

Of

course, we

willintroduce the

_definition

_of

$G$-operators soon, in the next section.

But

What exactly does

a

$G$-operator

mean

$Q$

(Q2) In Theorem 1, the value $1/\mathrm{q}$ in$f1(1/q)$

,

$\ldots$ :$f_{n}(1fq)$ looks

artificial.

Indeed, theradius

_of

convergence

_of

an

$E$

-function

is

infinite.

Onthe other hand, the radius

of

convergence

of

a$G$

-function

is

finite.

There is

a

difference,

of

course.

But Theorem 1 is not

_for

any points in the radius

_of

convergence, but only

_for

$1/\mathrm{q}$

.

It is the only

case

that the denorninator$q$ is (very) much larger than the numerator 1. We

are

afraid

that it is unavailable

for

technical

reasons.

Here,

we

think “is it good that$G$

-functions

are

regarded

as an

analogue to $E$

-functions

9”

So, the second question is

Find

a

betteranalogue.

From

now

on,

we

will consider the first question inthe next section,

Section

2, and

we

will attack the

\S 2

$G$-OPERATORS(AN _{ANSWER OF} $\mathrm{Q}$$1$)

In thissection,wegive the definition of$G$-operators. And weintroduce two results: Chudnovsky’s result

and Andre’s result. We

are

sure that these two results

are

one of the solutions of the first question in

the previous section. In addition,

we

recall the definition of $G$-operators; it is

an

iteration. We will also

introduce

a

slight generalization of$G$-operators. Then

we

will show

some

properties ofthem, and show

an

application. We think that it is a further consideration ofthe first question.

2-1 NOTATIONS AND DEFINITIONS

Let $v$ be a normalized valuation of$K$, the base number field, with the product formula, and such that

the absolute Height does not depend

on

the choice of thefield$K$

.

That is, if$K=\mathbb{Q}$,

$\{$

$|p|_{p}:=$ 1/p, padic valuation, ($\mathrm{p}$: prime)

$|\xi|_{\infty}:=|\xi|$ for$\xi\in \mathbb{Q}$, $|\cdot|$ isthe usual Archimedean valuation

The -adic valuation

is

this; for $m,n\in \mathbb{Z}$, $p^{e}||m$ ($p^{e}$ is the maximum

-factor

of

$m$),(i.e., $p^{e}|m$ and

$p^{e+1}\{m$),$p^{f}||n$, put $|rn/n|_{p}:=p^{f-e}$

.

The usual Archimedean valuation

means

that for $m/n$ $\in \mathbb{Q}$ with $m/n$ $\geq 0$, $|m/n|_{\infty}=m/n,$ otherwise,

$|m/n|_{\infty}=-m\prime n$

.

If

we

write$v\{\infty$,

we mean

that $v$ is non-Archimedean.

The product formula

means

$\prod_{v}|4|_{v}=1$ for $\xi\in K$

\

$\{0\},$where $v$in $\prod_{v}$

runs

every normalized valuation

of$K$

.

For $M=$(raij) $\in M_{n,n’}(K):n\cross n’$-matrix, every component is in$K$, weput $|M|_{v}:= \max_{i,\mathrm{j}}|m_{i,j}$$|v$

.

Foranyreal number $a\in \mathbb{R}$,

we

put$\log^{+}a:=\log\max(1, a)$

.

Definition, _{(sizes andglobal radii of powerseries)}

For$\mathrm{Y}=$ $\mathrm{i}$ _{$i=0\infty \mathrm{Y}_{i}xi\in M_{n,n’}(K[[x]])$},

we

put ’the size of$\mathrm{Y}’\sigma(\mathrm{Y})$

as

$\sigma(\mathrm{Y}):=\mu\overline{arrow}\mathrm{t}^{\mathrm{i}\mathrm{m}\sum_{v}}\infty$

;

$\max_{i\leq\mu}\log^{+}|\mathrm{Y}_{i}|_{v}$

$\in \mathbb{R}_{\geq 0}\cup\{\infty\}$.

Weput ’the global radii’ $\rho(\mathrm{Y})$

as

$\rho(\mathrm{Y}):=\sum_{v}\varlimsup_{\muarrow\infty}\frac{1}{\mu}\mathrm{m}_{i\mathrm{a}_{\mu}^{\mathrm{X}\log^{+}|\}_{i}|_{v}}}$ $\in \mathbb{R}\geq 0\cup$J$\{\infty\}$,

where $v$ in$\sum_{v}$

runs

over

every normalized valuation$v$ in K.

0

A paraphrase of the definition ofG-functions.

Every component

_of

$\mathrm{Y}=\sum_{i=0}^{\infty}\mathrm{Y}_{j}x^{t}\in M_{n,n’}(K[[x]])$

satisfies

the conditions (G-1) and (G-2) in the

definition

of

$G$-functions, is equivalent to, $\sigma(\mathrm{Y})<\infty$

.

$\square$

That is to say,

a

$G$-function is

a

solution of

a

linear differentialequation such that its size is finite.

We do not want to talk here abouttheradius of

convergence,

but

we are sure

thattheradius of

convergence

is a usual notion, but thesize is not

so

usual. So weremark ;

A relation between $\rho(\mathrm{Y})$ and the radius ofconvergence of Y. (Cf. [A])

Put$Rv\{Y$) $:=\varliminf_{iarrow\infty}|\mathrm{Y}_{t}|_{v}^{-1/i}$ (the radius

of

convergence at v), then $\rho(\mathrm{Y})=\sum_{v}\log^{+}\frac{1}{R_{v}(\mathrm{Y})}$

.

$\square$

88

It is obviousbythedefinitions, the

difference

between the size andthe global radiiisonlytheinterchange

of the limitation $\varlimsup_{\mu\prec\infty}$ andthe summation

$\sum_{v}$

.

Now,

we

havefinished defining the size and the global radii for power series.

Next, we define the size and the global radii for the lineardifferential equation (EQ).

Prom

now

on,

we

consider only

non-Archimedean

cases

$v\{\infty$

.

For

a

rational function $f\in K(x)$

, we

write the

s0-called Gauss’s

norm

_of$f$ at $v$

as

$|f|_{v}$

.

(That is, for

$f= \sum_{i=0}^{N}f_{i}x^{i}\in K[x]$, put $|f|_{v}:= \max_{i}|f_{i}|_{v}$, for $f,g\in K[x]$, $g\neq 0,$ put $|f/C$$|_{v}:=|f|_{v}/|g|_{v}$: well-defined.)

For$M=(m_{i,j})\in M_{n}(K(x))$: $\mathrm{n}\mathrm{x}\mathrm{n}$-matrix

over

the rational

function

field$K(x)$, put_{$|Ml|_{v}:= \max_{i,j}|m_{i,j}$}$|v$

.

Let I be the identity matrix.

Forthe matrix $A$ in (EQ),

we

consider the sequence $\{A_{i}\}_{i=0,1},\ldots$

4

$:= \frac{1}{i!}\mathrm{t}((\frac{d}{dx}+{}^{\mathrm{t}}A)^{\dot{*}}I)$ _{$\in M_{n}(K(x))$},

thatis, if

we

putthe map

as

$\varphi_{A}$ : $M_{n}(K(x))arrow M_{n}(K(x))$, $B_{r}*$ $(d/dx1 \mathrm{t}A|)$B,

then

${}^{\mathrm{t}}A_{1}$. $= \frac{1}{i!}\varphi_{A}\circ\cdot$

.

.

_{$\circ\varphi_{4}(I)$}

.

We note that this$A_{i}$ satisfies

$;$ $( \frac{d}{dx})^{i}m=A_{i}m$

for $m$in (EQ).

Definition. (sizes andglobal radii of linear differential equations)

We denote ’the size of(EQ) $\sigma(A)$’

as

$\sigma(A):=\mathrm{n}_{v}\mu\sum_{1\infty}\frac{1}{\mu}\max\log^{+}|A_{i}|_{v}i\leq\mu$ $\in \mathbb{R}_{\geq 0}\cup\{\infty\}$

.

We denote ’the global radii of(EQ) $\rho(A)$’

as

$\rho(A):=\sum_{v\{\infty}\varlimsup_{\muarrow\infty}\frac{1}{\mu}\max\log^{+}|i\leq\mu A_{i}|_{v}$ $\in \mathbb{R}\geq 0\mathrm{J}$$\{\infty\}$,

where

$v$ in$\sum_{v\{\infty}$

runs

over

every

normalized

non-Archimedean

valuation $v\{\infty$

of

K. Cl

Now

we

define G-0perators.

Definition. (G-0perators)

We call (EQ)

a

$G$-operator if$\mathrm{a}\{\mathrm{A}$) $<\infty$

.

Cl

(Thereis

no

way tocall$\mathrm{p}\{\mathrm{A}$)

as

finite.)

The definitions of$G$function and$G$-operators

are

similar, butthedifference isthat$G$-operators

use

the

2-2 CHUDNOVSKY’S RESULT

We think the following two theorems, Chudnovsky’s and Andre’sresults,show themeaningof G-0perators.

Theorem 2. ([C])

Under the assumptions

_of

Theorem 1, that is,

_for

$m={}^{\mathrm{t}}(f_{1}, \ldots, f_{n})$ in (EQ), $(n\geq 2)$, assume that

$f_{1}$,

$\ldots$ ,$f_{n}$ are $G$

-functions

and

assume

that $f_{1}$,

$\ldots$ ,$f_{n}$

are

linearly independent

over

$\mathbb{C}(x)$

.

Then $\sigma(A)<\infty$. that is, (EQ) is

a

$G$-operator. $\square$

It follows that theassumption, $”(\mathrm{E}\mathrm{Q})$ is

a

$G$-operator” in Theorem 1, is

unnecessary.

2-3 ANDR\’E’$\mathrm{S}$ RESULT

Theorem 2 (Chudnovsky’s result)says that “ifcomponentsof$m={}^{\mathrm{t}}(f_{1}, \ldots, f_{n})$

are

linearlyindependent

over

$\mathbb{C}(x)$, then $\mathrm{a}(\mathrm{m})<$

oo

implies $\mathrm{c}\mathrm{r}(\mathrm{A})<\infty$.”

The next theorem,

a

result ofAndre, says the opposite ofthis Theorem 2in

some sense.

We

now

consider the “matrix solution” of (EQ).

$\frac{d}{dx}X=AX,$ $A\in M_{n}(K(x))$

Here

we assume

that:

there exists

3

$C\in M_{n}(K)$, there exists$\exists 1$$\mathrm{Y}=$ $\mathrm{E}7\infty=0\mathrm{Y}_{i}x^{i}\in$ Mn$\{\mathrm{K}[[\mathrm{x}\}])$ with$\mathrm{Y}_{0}=I$ (theinitialcondition)

such that

$X=i’ x^{C}$ $(= \mathrm{Y}\sum_{i=0}^{\infty}\frac{1}{i!}(C\log x)^{i})$

’plus’ somesuitable conditions.

Then

Theorem 3. ([A])

Undersome conditions,$\sigma(A)<\infty$ implies$\mathrm{a}(\mathrm{m})<\infty$

.

$\square$

Finally, under “some conditions,” $G$-functions and $G$-operators

are

equivalent (in the meaning).

This is exactlywhat a $G$-operator

means.

Butwe go on to

a

further consideration.

2-4 A GENERALIZATION OF $G$-OPERATORS

Let’srecall the definition of$G$-operators. It

comes

from the iteration ofthe map:

$\varphi A$ : Mn$(\mathrm{K}(\mathrm{x}))arrow \mathrm{M}\mathrm{n}(\mathrm{K}(\mathrm{x}))$

.

This map defines

a

G-0perator

So

now

we

consider the followingsequence:

Let $T$,$A$,$B\in$Mn$(\mathrm{K}(\mathrm{x}))$

.

We define the sequence $\{(T, A, B)^{\langle i\rangle}\}:=0,1,\ldots\subset M_{n}(K(x))$

as

$(T, A, B)^{(0\rangle}:=T,$

$(T, A, B)^{(i+1\rangle}:= \frac{1}{i+1}(\frac{d}{dx}(T, A, B)^{\langle i\rangle}-\{(\mathrm{T}, A, B)^{\langle:\rangle}+(T, A, B)^{\langle i\rangle}B)$

We put

$\sigma(T, A, B):=\varlimsup_{\muarrow\infty}\sum_{v\{\infty}\frac{1}{\mu}\max_{i\leq\mu}\log^{+}|(T, A, B)^{\langle i+1\rangle}|_{v}$, $\in \mathbb{R}\geq\circ\cup\{\infty\}$,

$\{(\mathrm{T}, A, B):=$ $1$ $\varlimsup_{\muarrow\infty}\frac{1}{\mu}\max_{\dot{l}\leq\mu}\log^{+}|(T, A, B)^{\langle i+1)}|_{v}$, $\in \mathbb{R}\geq 0\cup\{\infty\}$

.

$v\{\infty$

88

Proposition 4. ([Nl, N2])

Let$T$,$T_{1}$,$T_{2}\in \mathrm{G}\mathrm{L}_{n}(K(x))$, andlet$A$,$B$,_{$C\in M_{n}(K(x))$}, Then

(0) $\mathrm{r}(A)$ _{$=\sigma(I, 0, A)$}.

(1) $\sigma(I, A, A)=0.$

(2) $\sigma(T, A, B)=\sigma(I, 4, T[B])=\sigma(I, T^{-1}[A], B)$, where $T[B]:=TBT^{-1}+((d/dx)T)T^{-1}$

.

(3) $\sigma(T_{1}, A, \mathrm{O})=$a(T2,0,$A$).

(4) $\sigma(T_{1}T_{2}, A, B)\leq\sigma(T_{1}, A, C)+\sigma(T_{2}, C, B)$

.

(triangle inequality)

For$\rho$, the

same

(in)equalities hold. $\square$

Theideaofthis proof is : first, show

some

equationsforthesequences byinduction, _{then,apply them to}

the

definition

to$\sigma$

.

Thetriangle inequality (4)

comes

fromthe triangleinequalitiesof valuations. Q.E.D.

Although it is

a

good opportunityto remark about Proposition4 here,

we

would like to do it in thelast

section, Section 4.

As

an

application of Proposition 4,

we

obtainthe nexttheorem.

Let’s recall Theorem 2, Chudnovsky’s result. Chudnovsky’s result

assumes

the linearly independenceof

$G$-functions. And it is for

a

vectorpower seriessolution. Then,

one

might think it is not the strict contrary

ofAndr\’e’s result, Theorem 3. Here

we

obtain ;

Theorem 5. ([N2])

Under the assumptions

_of

Theorem 3, $\sigma(\mathrm{Y})<$

oo

implies$\sigma(A)<\infty$

.

$\square$

We

can

say that these resultsinthis sectionis

an

answer

of Question 1 in

Section

1.

fi3

$G$_{-FUNCTIONS AND DIOPHANTINE APPROXIMATIONS, REVISITED} (AN _{ANSWER OF Q2)}

In this section,

we

want to consider the next question Q2: “Find

a

betteranalogue for G-functions”

The main idea in thissection is that $” G$-functions

are

not

so

similar to $E$-functions, but algebraic

func-tions”. We would like to presumethis

a

candidate of thesolutions of the second question in Section 1. So

we

will introducehere

a

“circumstantialevidence.”

First

ofall,the author has

an

easy thought.

For instance, the valuesof algebraic

functions

over

rationals at algebraic pointsare, of course, algebraic

numbers. Recall that algebraic functions$/\mathbb{Q}$

are

also$G$-functions. And alsorecall,Theorem$\mathrm{E}$isan analogue

to Lindemann’s Theorem, which is

a

point of view from the transcendental number theory. So,

one

may

think, it is hard to regard the problemsofspecial valuesof$G$-functions likeas theonesofE-functions.

So, just in aprivate opinion, $G$-functions

are

not so similar to$E$-functions, but

as

algebraicfunctions.

This is

a

private solution of Question 2: $” G$-functions is analogueto algebraicfunctions.”

In thissection,

we

introduce

a

“circumstantialevidenc\"e’’ with this opinion.

3-0

ALGEBRAIC FUNCTIONS

We recall algebraic functions’

case

But

we

don’t knowthe “smart formulation”, ,

so

we

are

afraid that

we

might not give

a

goodexplanation.

Proposition 6. (Liouville’s inequality)

(We _{consider only the genus}

₀

case, and

we

restrict

a

special case.)

Put

$5_{1}:=\{x\in K|\exists y\in Ks.t. f(x,y)=0\}$

$=\{g(y)\in K|y\in K\}$

.

Let$t\in K$ with $\frac{d}{dy}g(t)\neq 0.$ Put$a:=g(t)$

.

Then there exists$\exists c>0$ such that

$|$

a

$-a|> \frac{c}{H(\alpha)^{[K.\mathrm{Q}]/n}}$.

for

$\forall\alpha\in S_{1}$ with$\alpha\neq a.$

Here $H(\alpha)$

means

the absoluteHeight

of

$\alpha$

.

$(i.e., \mathrm{H}(\mathrm{a})=$ $\sum v\log^{+}|\alpha|_{v}.,)$ ($c$ is independent

of

$\alpha$.) Cl

It is the s0-called Liouville’s inequality if$g(y)=y.$

We know also that “there exists

an

estimate of the number of rational points of irreducible algebraic

curve

$/\mathbb{Q}$in generalcases”, but

we

will mention it soon.

Anyway, that isto say, for the “rationalpoints” $S_{1}$, the inequalityholds. This is Proposition 6.

3-1

$G$-FUNCTIONS

For $G$-functions,

we

obtain

an

analogueto Proposition6.

Theorem 7. ([N4,N5])

Let $\zeta_{0}\in K$ be given. Put $D\subset \mathbb{C}$:

a

closed disc centered $\zeta_{0}$ with radius $<1/2.$ Let $\mathrm{d}\{\mathrm{x}$) $\in \mathrm{Z}[\mathrm{x}]$ be the

common

denominator

_of

all components

_of

$A$ in (EQ).

We assume that

(1) (EQ) is

a

$G$-operator. $n\geq 2.$

(2) $m={}^{\mathrm{t}}(f_{1}, \ldots, fn)$ is analytic on$D$.

(3) $f_{1}$,

$\ldots$ ,$f_{n}$ are linearly independent

over

$\mathbb{C}(x)$

.

44) $\{z\in \mathbb{C}|d(z)=0\}$ I $D=\emptyset$

.

Put

$S_{K}:=\{(\in D\cap K|\exists\kappa_{\zeta}\in \mathbb{C}\mathrm{Z}\{0\}s. t. \kappa\zeta m(\zeta)\in K^{n}\}$

.

Then

we

have

(i) (For$\zeta_{0}\in S_{K}$ and)

for

any

small real $\mathrm{V}$ _{$\epsilon>0,$} there exists

3

$c<\infty$ such that

$|$(

$- \zeta_{0}|>\frac{1}{H(\zeta)^{[K\mathrm{Q}](_{n}^{1}+\epsilon)}}$

for

$\forall$ (_{$\in S_{K}$} with

$\mathrm{H}(\mathrm{a})\geq c$

.

Here $c$ depends

on

$H(\zeta 0)$,$A$,$\epsilon$,$D$, and is independent

of

$\langle$.

(ii)

$\varlimsup_{Barrow 0}\frac{\log\neq\{\zeta\in S_{K}|H(\zeta)\leq B\}}{1\mathrm{o}\mathrm{g}B}\leq\frac{4}{n}[K : \mathbb{Q}]$

.

(the trivial estimate $is\leq 2[K:\mathbb{Q}]$) $\square$

Remark8.

(1) For algebraic functions’ case,

a

similar estimate in (ii) with $\leq\frac{2}{n}[K : \mathbb{Q}]$ holds (for $K=\mathbb{Q}$ with

different notations). See [Se] for the details.

(2) If$f_{1}$,

$\ldots$ ,$f_{n}$

are

algebraically independent

over

$\mathbb{C}(x)$,

we

have

.

_the

_left

hand sidein (i) $\geq 1/H(\zeta)^{\epsilon}$ holds.

.

the left hand side in (ii) $=0$ holds. $\square$

One seesthat Theorem 7is similar to Proposition6 and the first remark (1) in Remark 8.

Moreover, Theorem 7does not containartificial conditions like

as

$1/\mathrm{g}$in Question 2.

In conclusion, these mightbeananalogue of

an

appearance. But

even

so,

one

canthinkwe may saythat

the similarity between Proposition 6 and Theorem 7 gives acandidate of the solutions of Question 2. It

80

\S 4

FURTHER PROBLEMS

We would like to introduce

some

further problems.

First,

as

for G-0perators:

(Q3) We know the triangleinequality in Proposition 4 (4).

“Find somethinginteresting related to Proposition 4and develop it.” $\square$

The author has spent much much time

on

it. But invain. No results (except Theorem5).

Here

we

would like to remarkabout Proposition4:

Remark.

(1) Ifwe put

$\mathrm{a}(\mathrm{A}, B):=\frac{1}{2}$(a ($I$,$A$,_$B)+$a(A,$B$,$A)$)

then from (1) and (4) in Proposition 4, the map

$\overline{\sigma}$

:

$M_{n}(K(x))\mathrm{x}$ $M_{n}(K(x))arrow \mathrm{R}_{\geq 0}\cup\{\infty\}$

is

a

pseudodistancefunction, thatis, $\overline{\sigma}(A, A)$$=0,\tilde{\sigma}(A, B)$ $=\overline{\sigma}(B, A),$ $\sigma-(A, B)$ $\leq\overline{\sigma}(A, C)$$+\overline{\sigma}(C, B)$

.

But...

(2)

we

don’t know any relation between $A$ and $B$ with $\sigma(I, A, B)=0.$

(3) furthermore, for any $A$ and $B$

, we

don’t know whether$\sigma(I, A, B)=$ a(A,$B,$$A$) holds

or

not.

It

means

that,

a

pseudodistance function$\tilde{\sigma}$

on

$M_{n}(K(x))$

are defined

byProposition 4, but, that’sall

we

can

do. $\square$

Next,

as

for Diophantine approximations and

G-functions.

(Q4) Theorem 7is,

one may

think, Liouville’s inequality for G-functions.

So, “Does a Roth’s typepropertyon $G$-functions hold?” $\square$

Here the original

Roth’s theorem

is ;

(Roth’s theorem) ([R]): For

a

given $\alpha\in\overline{\mathbb{Q}}$with_$\deg$

ce

$\geq 3$ and for any real $\epsilon>0,$

$| \alpha-\frac{p}{q}|>\frac{1}{q^{2+\epsilon}}$

holds

for any

$p$,$q\in \mathbb{Z},$ $\neq 0,$with $(p, q)=1$ but finite exceptions.

0

Firstofall,the author has no idea even how to formulate it for$G$-functions. And he thinks that it is not

so easy, probably verydifficult.

Forinstance, the proofof theoriginal Liouville’s inequality requires only

a

few pages, probably, only

one

page, butthe proofofthe Roth’stheorem requires 20 pages. Thismeans, the number of pages of the proof

of the Roth’s theorem is

20

timesthan the

one

ofLiouville’s inequality.

Here, now, Theorem

7 Liouville’s

inequality

for

$G$-functions, needs

more

than

30 pages. Then

IF

one

could prove any Roth’s type properties for $G$-functions, itwould need, probably, 20

times 30

equals to

600

pages!?

Of course, it isjust

a

joke,

a

largestory. Butinthis stuation,

we

can

notbelievethat anyproofsof Roth’s

typestatements do not requireanycomplicated arguments.

(Q5) It is well known that the number of$\mathbb{Q}$-rational points of any (projective, smooth) algebraic

curve

withlargegenusisfinite.

HowaboutG-functions? Forexample,how about thefiniteness of thenumberof the set$S_{K}$ inTheorem 7?

“Especially, inTheorem

7

if$f_{1}$,

.

. .

’$f_{n}$

are

algebraically independent

over

$\mathbb{C}(x)$, does $\mathit{1}S_{K}$ $<$

oo

hold?” $\square$

(Q6) The special value of the polylogrithmic function$L_{k}(x)$ at $x=1$, $Lk())$ equals to $\zeta(k)$ (theRiemann

zeta-function) for $k\in \mathbb{Z}_{\geq 2}$.

“Apply$G$-functiontheoryand deduce something about the special values ofzeta-functions.”

a

This is “a bigproblem.”

In the present situation, $G$-function theory tells NOTHING about special values at any points

on

the

radius ofconvergence.

$\zeta(k)$ is the special value ofthepolylogarithmic function $L_{k}(x)$ at the point $x=1,$ it is on the radiusof

convergence ofthepolylogarithmic function Lk(x). Therefore the present $G$-functiontheory

can

tell nothing

about it in the presentsituaton.

The reader maythink that every problem is, easy to say, but difficult to carry it out.

As for the number theory, the last question Q6 is very attractive, butwe almost give uptheQuestion6

without someinnovation.

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OIWAKE-CHO SAKYO-KU Kyoto, 606-8502, JAPAN