On the ultrametric Stone-Weierstrass theorem and Mahler's expansion

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

AUL

-J

EAN

C

AHEN

J

EAN

-L

UC

C

HABERT

On the ultrametric Stone-Weierstrass theorem

and Mahler’s expansion

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 43-57

<http://www.numdam.org/item?id=JTNB_2002__14_1_43_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

On the ultrametric Stone-Weierstrass theorem

and

Mahler’s

_expansion

par PAUL-JEAN CAHEN et JEAN-LUC CHABERT

RÉSUMÉ. Nous

_explicitons

une version

ultramétrique

du théorème

de Stone-Weierstrass. Pour une

_partie

E d’un anneau de valuation V de hauteur

_1,

nous _montrons, sans aucune

hypothèse

sur le corps

résiduel,

que l’ensemble des fonctions

polynomiales

est dense dans

l’anneau des fonctions continues de E dans V si et seulement si la clôture

_topologique

Ê

de E dans le

_complété

de V est _compacte. Nous

_explicitons

ainsi le

_{développement}

d’une fonction continue en série de fonctions

_{polynomiales.}

ABSTRACT. We describe an ultrametric version of the

Stone-Weierstrass

_theorem,

without _any

_assumption

on the residue field.

If E is a subset of a rank-one valuation domain

_V,

we show that

the

_ring

of

_polynomial

functions is dense in the

_ring

of continuous

functions from E to V if and

_only

if the

_topological

closure Ê

of E in the

_completion

of V is compact. We then show how to

expand

continuous functions in sums of

_polynomials.

Introduction

The classical Stone-Weierstrass theorem says that the

ring

of real

polynomial

functions is dense in the

_ring

of real continuous func-tions on a

_compact

subset E of endowed with the uniform convergence

topology. Obviously,

Q[X]

l

is then also dense in

Dieudonn6

_proved

a similar

p-adic

result

[9,

Theorem

4],

replacing

R

by

the

_p-adic

_completion

_~

of

_Q,

and this was extended

_by

Kaplansky

[13]

to

a

_compact

subset E of a valued field L for a rank-one valuation

(with

no

assumption

on the residue

field) (see

also

[12,

Theorem

32]).

Since the

_ring

of

_{p-adic integers}

_Zp

is

_compact,

then every

function 0

E

_Zp)

_may

be

_{uniformly approximated by polynomials}

in

_and,

in

_particular,

the

ring

Int(Z)

E

_Q[X]

of

_{integer-valued polynomials}

is dense in the

_ring

Considering

a domain

D,

with

quotient

field

K,

and a subset E of

_K,

we more

_generally

introduce the

ring Int(E, D)

of

_{integer-valued polynomials}

on E:

In

_particular,

for E =

D,

_we let

Int(D) = { f

_E

K[X]

I f (D)

g

D}

be the

_ring

of

_{integer-valued polynomials}

on D. If V is a discrete valuation

domain with finite residue

field,

its

_completion

V is

_compact,

and

_hence,

the

polynomial ring

Int(V)

is dense in the

_ring

of continuous functions

_{C(V, V),}

for the uniform convergence

topology

[4,

Theorem

_{111.3.4] (in}

_fact,

one _may even

replace

V

by

a

local,

one-dimensional Noetherian domain

D,

with

finite residue field

_D/m,

such that D is

_{analytically irreducible,}

that

_is,

its

completion

D

in the m-adic

_topology

is a domain

[4,

Corollary

111.5.4]).

Considering

a subset E of a discrete valuation domain V such that E

is

_compact,

it follows from

_{Kaplansky’s}

extension of the Stone-Weierstrass

theorem that the

_{polynomial ring}

_Int(E,

_V)

is dense in the

_ring

C(2?, V)

of continuous functions from

2?

into

V,

with no

_assumption

on the residue

field

_(and

we showed

recently that,

here

again,

we could

replace

V

by

a

local,

one-dimensional Noetherian domain which is

_analytically

irreducible

[5,

Proposition

4.3~).

On the other

hand,

Mahler

_[14,

Theorem

_1]

_gave an

explicit description

of the

_expansion,

of a continuous

function

E

_Zp),

in series of the

form

where

For the domain V of a discrete valuation v with finite residue

_{field, replacing}

the _sequence of

_{integers by}

a

_’very

well distributed’ _sequence and

the binomial

_polynomials

_{( ~ )}

_by

the

_{integer-valued polynomials}

_{f n (X) _}

11’

_k=O

_X,

_un Amice

_[1,

_Chap.

_II]

_generalized

this

_description

to the

expan--ui, 1

sion of a continuous

function 0

E

_{C(V, V )}

_(see

also

_Wagner

_[21]

for

_positive

characteristic) : §

can

uniquely

be

expanded

in the form

Moreover

It is then said that the

_{f n’s}

form a normal basis of the ultrametric Banach

space

C(V, K)

~1~.

In

_fact,

Amice’s result

_applies

more

_generally

to

_{functions 0}

E

_C(E,

V )

on

’regular’

_compact

subspaces E

of

V.

In a recent paper,

Bhargava

and

Kedlaya

[3]

extended Amice’s results to _every

_compact

subset E of

_V,

using

the notion of

_{’v-ordering’,}

introduced in

_[2],

which _appears to be a

fine

_{generalization}

of Amice’s very well distributed sequences. An alternate

version of this result

_appeared

also

_recently

_[18,

Theorem

_1.2],

where the

sequence is

replaced by

the sequence of the Fermat

polynomials,

which are known to form a basis

of Int ( V) ([10]

or

[4,

Proposition

11.2.12]).)

Despite

their

_{generalizations}

to

_subsets,

these

expansions

in series con-cern

_only

the case of a rank-one discrete valuation domain with finite residue

field, although

the Stone-Weierstrass theorem holds without this finiteness

hypothesis, and,

in

_fact,

also for a non discrete rank-one valuation domain.

The aim of this _paper is to

_give

an extension of these results in this more

general setting,

and in the same

_time,

to

_{provide proofs}

which are

significantly

shorter than in

_[1]

and

_[3]

and also more

elementary

than in

papers

dealing

with normal bases such as

[20].

Moreover we

conversely

establish that the

_compactness

of

E

is _necessary in the ultrametric Stone-Weierstrass theorem.

HYPOTHESIS: We let V be the

_ring

of a rank-one valuation _v, with

quotient

field

_K,

and we consider a subset E of K. We denote

by V,

K and

E

the

_completions

of

_V,

K and E

_(but

_simply

denote

_{by v}

the extension of the valuation

to K).

For sake of

_{completeness,}

we first establish an

_analogue

of the

Stone-Weierstrass theorem

_which,

in

_fact,

can be derived from

_{Kaplansky’s}

_general

results

_[12,

Theorem

_32]:

if

E

is

_compact,

the

_ring

_{Int(E, V)}

of

integer-valued

_polynomials

is dense in the

_ring

C(2?, V)

of continuous functions from

E

to

V.

In the second

_section,

_assuming

moreover the subset E to be

infinite,

we use this result to

_{give explicit}

series

_expansions

of continuous functions.

_Finally,

in the last

_section,

we _prove the converse: the

Stone-Weierstrass theorem holds if and

_only

if the

_completion

E

of E is

_compact.

1. The Stone-Weierstrass theorem

Let us recall that the absolute value of x is defined

_by

Ixl

= thus

K is an ultrametric _space, endowed with the distance

For a E R and a E

_V,

we denote

by

Ba (a)

the closed ball with center a and

radius e-a :

The balls with center a form a fundamental

_system

of

clopen neighborhoods

of a.

If

_f

E

_K(X~

is a

polynomial

with coefficients in

K,

and b a common

denominator of its

_{coefficients,}

then

_{b f}

E

_V(X~,

and

_hence,

for _{x, y} E

_V,

we

that

_is,

_f

is

_uniformly

continuous on V.

Now,

if E is a

_compact

subset of

K,

then E is

_bounded,

that

_is,

_lxl

_M,

for each x E

_E,

or

equivalently,

E

is a

_fractional

subset of

_V,

that

_is,

there is a nonzero element d E V such that

Under the

_hypothesis

that E is a fractional

_{subset, letting g}

=

f (X/d),

we have

f (x)

=

g(dx),

and it follows that

f

is

uniformly

continuous on E.

We can thus consider

K[X]

as a

_subring

of the

_{ring C(E,}

K)

of continuous

functions from E to

_{K, similarly}

we can consider the

_ring

Int(E, V)

of

integer-valued polynomials

as a

subring

of the

ring

C(E, V)

of continuous

functions from E to V.

We wish _{to prove}

_that,

if E is

_compact,

then

_{Int(E, V)}

is dense in

C(E, V).

First we establish that the

polynomials

in

Int(E, V)

_separate

the

points

of E

_(a

_property

similar to that of

_{interpolation}

domaans

_[6]):

Lemma 1.1. Let K be the

_quotient

_{field of}

a rank-one valuation domain

V and E be a

_compact

subset

of

K. For each

_{pair of}

_points

_a,

b E E,

there

exists

_f

E

_{Int(E, V)}

such that

_{f (a)}

= 1 and

_{f (b)}

= 0.

Proof.

Set

_{v(a - b)}

=

q. Since E is

compact,

it is contained in the finite

union of

_disjoint

closed balls

_Ry(6t),

with centers

_{ba, bl, ... , bn,}

one of them

containing

a and b. With no loss of

_generality,

we set

_bo

= b. We _prove,

by

induction on _n, that there is a

polynomial

_g, with coefficients in

_K,

such

that

_g(b)

= 0 and

v(g(x)) &#x3E; v(g(a))

for all x _E E.

Hence,

the lemma is

proved,

with

_f(X)

= -

For one

ball,

that

is,

E C

B.y (b),

we have

v (x - b) &#x3E; v (a - b),

for each x E E. We can thus

_{take g}

= X - b.

- Assume the result

to be true for n balls and consider the case of n + 1

balls: E c We set

_ði

=

v(b -

bi)

and order

bl, b2, ...

bn

in such a _way

that

b2 &#x3E; ... &#x3E; 6n.

We note

_{that q}

&#x3E;

_61,

and

_hence,

v(a-bn)

=

6n;

_{we note} also

that,

for 1 i

n,

6n -

We let

El

be

the intersection of E with

_Uoan-1

_By

induction

_hypothesis,

there is a

polynomial

_gl such that

91 (b)

= 0 and

v ~gl (x)) &#x3E; v (gi (a))

for each

x E

_El.

_Multiplying

gi

by

a

_constant,

we _may assume that _gi E

_Int(E,

_V),

and

_hence,

that

_{v (gl (x)) &#x3E;}

_0,

for each x E E. Since the value _group of the valuation is a

subgroup

of

Jae,

and

(’Y -

6n)

&#x3E;

_0,

there is an

integer

m

such that

_{m(’Y - b~) &#x3E;}

We claim that we can

_{take g}

of the form

9 =

gi (X - bn) m.

For

this,

we consider two cases:

o If x

_{E El,}

then x _E

B’Y(bi)

for some i n. Then

v(x-bn)

=

6n-Thus,

we have

but z g

El,

we have x E that

_is,

_~y. It follows

We are

_ready

for a first version of the Stone-Weierstrass theorem

(the

proof

is similar to

_[4,

Exercise

_III.20~):

Lemma 1.2. Let K be the

_quotient

_{field of}

a rank-one valuation domain

V and E be ca

_compact

subset

of

K. Then

Int(E,

V)

is dense in the

ring

C(E, V)

of

continuous

_{functions from}

E to

_V,

endowed with the

_uniform

convergence

topology.

Proof.

Each continuous function can be

arbitrarily approximated by

a

lo-cally

constant function. Thus it is

_enough

to prove

that,

for each

clopen

set

U,

the characteristic

function 0

of U can be

_approximated

_by

an

integer-valued

_polynomial:

for each A &#x3E;

_0,

there exists a

polynomial f

E

_{Int(E, V),}

such that

_{v ~ f (x) - ~(~)~ &#x3E; A,}

for each x E E. Choose a E U. For each

b ft

U, by

the

_previous

_lemma,

there is a

_{polynomial fb}

E

_{Int(E, V),}

such that

_fb(a)

=

_1,

and

fb(b)

= 0. Since

fb

is

continuous,

we have A

for each x in some

_neighborhood

of b. Since U is a

clopen

set and E is

compact,

there is a

_product

_ga of a finite number of such

_polynomials

such

that

_ga(a)

=

1,

and A for each _x in the

complement

of U in E.

Since _{ga is}

_continuous,

we have

v (ga (X) - 1) ~!

A,

for each x in some

neigh-borhood of a. Since U is

_compact,

there is a finite

_product

fl (1 - ga)

=

1-g

with such

_{polynomials ga}

such that

_{v~g(x) - 1~ &#x3E;}

_A,

for each x E U and

v (g(z)) &#x3E;

A,

for each x E

_{E B}

U. D

In

_fact,

it is often the case that the

topological

closure #

of E in the

completion

V

of V is

_compact

while E is not

_(as

for the

_ring

V

_itself,

if V is a -

non-complete

-

rank-one discrete valuation

_domain,

with finite residue

_field).

If

E

is

_compact,

then E is a fractional subset

of V,

the

polynomials

with coefficients in K are

_uniformly

continuous on

E,

thus

Int(E, V)

can be considered as a

_subring

of the

_ring

C (Ê, V)

of

(uniformly)

continuous functions from

k

to

V,

and

_K[X]

as a

_subring

of the

_ring

C(E, K)

of

_(uniformly)

continuous functions from E to

K.

We thus derive another version of the Stone-Weierstrass theorem:

Proposition

1.3. Let K be the

_quotient

_{field of}

a rank-one valuation

do-main V and E be a subset

_of

K such that

P

is

_compact.

_Then,

(i) Int(E,

V) is

dense in

C(E,V)

_for

the

_uniform

_convergence

_topology,

(ii)

K[X]

is dense in

C(.9, K)

_for

the

_uniform

_convergence

_topology.

Proof.

(i)

From the

_{previous lemma,}

Int(E,V)

is dense in

C (f, f7):

each

4&#x3E;

E

_{C(#, V)}

can

_arbitrarily

be

approximated by polynomials (with

coeffi-cients in

K).

In

_fact,

as

K[X]

is dense in

each 0

E

C(Ê,

_V)

can be

is such that

_{v ( f (x) - ~(x) &#x3E;}

0 for every x E

Ê,

then

_f

is

_clearly

an

integer-valued polynomial.

(ii)

Since.9

is

_compact,

each

_{function §}

E

C (#, K)

is bounded.

_Multiplying

~ by

a

_constant,

we obtain a function in

C(E, V).

Hence the result follows

from

_(i).

D

2. Mahler’s

_expansion

We assume from now on that E is an infinite subset of K. We first

generalize

the notion of introduced

_by

_Bhargava

for a subset of a discrete valuation domain.

Definition 1. Let v be a valuation on a field

~,

and E be a infinite subset

of .K. We _say that a _sequence

funInEI4

of elements of E is a

_v-ordering

of

E

_if,

for each n &#x3E;

_0,

and each x E

_E,

we have

It is immediate that such an

(infinite)

_sequence exists in the case where v is rank-one discrete. We show that this is also the _case, for a rank-one

valuation,

if

E

is

_compact:

Lemma 2.1. Let v be a rank- one valuation on a

field

K,

and E be a

infinite

subset

_of

K such that

E is

_compact.

_Then,

_for

each a E

_E,

there exzsts a

v-orde7ing

funInEI4

of E

such that _uo = a.

Proof.

The _sequence is obtained

_inductively.

_Supposing

we have

obtained the first n

_elements,

we want to find un which is a minimum for the

continuous function

_{T (x)}

=

v (H n-1 u~) .

Since

E

is

_compact,

such a

minimum is reached for _{some yn} E E. On the other

_hand,

if un E E is close

enough

to yn

(to

be

precise,

if

_{v(un - yn)}

&#x3E;

_{v(yn -}

_Uk),

for 0

_{k n -1),}

then D

We now _suppose, as in the first

section,

that V is a rank-one valuation

domain

_{(corresponding}

to the valuation

_v).

Given a

_v-ordering

~un~n

of

the subset E of

_K,

we set

It is then _easy to show that is a

_regular

basis of the V-module

Int (E, V),

that

_is,

a basis such that

In

is of

degree n

for each n

[2,

Theorem

1].

Indeed,

it follows from the definition of a

_v-ordering

that each

_{f n}

is in

Int(E, V),

and that

_they

form a basis follows from the fact

_that,

for each

By analogy

with real

_analysis,

the numerators of the

_{f n’s,}

that

_is,

the

polynomials g~ (X ) -

_{fl§~I§ (X -}

Uk)

may be

called

_{Tcheb ychev}

polyno-mials

_[11].

_Indeed,

for the norm of the uniform _convergence on

E,

one

has:

But, contrarily

to the real _case, there is no

_uniqueness

for the

_Tchebychev

polynomials

since there is no

_uniqueness

for the

_v-orderings.

We shall _prove that each continuous function can be

_expanded

in series

of the

_{form 0 =}

From the Stone-Weierstrass

_theorem,

we first

show that it is

_always

_possible

to

_expand

continuous functions in series of

integer-valued polynomials.

For sake of

_completeness

we

_give

the

_proof

of

the next lemma that we _may

find,

for

instance,

in

[19,

Lemma

1.2].

Lemma 2.2. Let K be the

_quotient

_{field of}

a rank-one valuation domain V

and E be a subset

_of

K. Denote

_by

t an element

of

V such that

v(t)

&#x3E; 0.

_If

E is

_compact,

then _every continuous E

_{V )}

can be

_expanded

in the

_{with gn e Int(E, V).}

Proof.

Since

_{Int (E,}

_V)

is dense in

C(E, V )

_[Proposition

_1.3],

there is a

poly-nomial _go E

_{Int (E,}

_V)

such

_that,

for all x E

P, v (O(x) - go (~) ) &#x3E;

_{v (t) .}

That

is, 0

= _go

+ t~1,

where

~1

_E

C (E,

V ) .

Similarly,

one can find _gi E

_{Int (E, V)}

and

_{02 E}

V ),

such that

₁

= ₉₁ +

t02,

and

hence, 0

=

go +

tgl

+

t2 2

-Proceeding by induction,

we obtain the desired

_expansion.

0

We next show how to obtain a series

_expansion

from another one.

Lemma 2.3.

_{Let ~ -}

_{cngn, be a}

series

_expansion

_of

a continuous

function 0,

such that each

_{9n is}

an

integer-valued pol ynomial

on

E,

and

with

_coefficients

cn E K such that lim cn = 0. Consider a set

n-+oo

of

generators

of

the V -module

_{Int(E, V),}

and

_decompose

each _gn on these

generators: 9n =

bk,nvk,

with

_bk,n

E V.

_{Then, for}

each

_{k E N,}

the

series bk

=

E’o

cnbk,n

converges in

V, lim bk

=

0,

bncpn.

n=

k--oo

-Proof.

The series

_{Cnbk n}

_converges, since lim c, = 0. Fix N E

_N,

, n-+oo

there

_{is j}

E N such that n

_{&#x3E; j implies}

_{v (cn ) &#x3E;}

N. Let =

sup{ io,... ,

For k &#x3E;

_{i ( j ),}

we then have

_cnbk,n,

and

_hence,

v (bk ) &#x3E;

N.

Therefore,

lim

_bk

=

0,

and the series

bnpn

converges.

Now,

for each

k--oo

-j,

consider the difference

From the definition of the coefficients

_b~,

we have

And

_hence,

N,

that

is,

lim = 0. In other

words,

nfEfl

We then obtain the

_{following, generalizing}

_[3,

Theorem

_{1~ :}

Theorem 2.4. Let K be the

_{quotient field of}

a rank-one valuation domain

V and E be an

in finzte

subset

of

K such that

E

is

_compact.

Considering

a

v-ordering

funInEN

of E,

and

letting

fn(X) =

X-,

then _every

Un-U,l k

continuous

_{function 0}

E

C(.9, K)

can be

_expanded

in a series

_of

the

_form

~

=

an f n,

with lim

an = 0. The

coefficients

an E

K

are

uniquely

- _n~oo

determined

_by

the recursive

_formulae:

Moreover

Proof.

The

_possibility

to

_expand

a

function 0

E in a series of

the

_{form 0 =}

with

_{lim an = 0,}

follows

_immediately

from

- _n-&#x3E;oo

Lemma 2.2 and Lemma 2.3. This

_generalizes

to a

_{function 0}

since 2

is

_{compact, 0}

is

_bounded,

and

_hence,

C (E, V)

for some nonzero

constant a E V.

The recursive

_formulae,

follow

_immediately

from the fact that

_!n(un)

= 1

and

_fk(un)

= 0 for k _{&#x3E; n.} In

particular,

the

ak’s

_are

uniquely

determined.

Finally,

let a =

infnEN

v(an);

as

v(an)

-~-~ _oo, this infimum is reached.

Let no be the smallest

integer

such that

v(ano)

= a. From the recursive

formulae,

we have

We conclude that we have

_equalities

from the

_following

(obvious)

lemma.

a Lemma 2.5.

_Let

be the sum

_of

a

series,

where

a _sequence

_{of integer-valued polynomials,}

and where the

_coefficients

_cn

E K

are such that

lim cn

= 0. Then we have

n

that is, sUPxEÊ

IØ(x)1

sUPnEN

Remark 2.6. The

_previous

result may be

interpreted

in the

theory

of

normal bases. Let M be an ultrametric Banach _space on K and let

Mo

be

the unit

_ball,

that

_is,

the

sub-V-module

_Mo

=

~x

_E M

1}.

Recall

that a

family

lei I i E I }

of elements of

Mo

is called a normal basis

[1]

or an orthonormal basis

([16]

or

[17])

of M if each x E M has a

_unique

representation

of the form x =

EIEI

_xaez with

xi E K,

lim,

0 where

F denotes the filter formed

by

the cofinite subsets of

_I,

and

_Ilxll

_

¿iEllxil.

(i)

Of course, the

polynomials {fn}nEN

in Theorem 2.4 form a normal

basis of the ultrametric Banach space

C(#, K).

(ii)

The theoretical existence of such a normal

basisofc (f, K)

with a

polynomial

of each

_degree

was

_{already proved by}

Van der Put

[20,

_Prop.

5.3~,

however these

_polynomials

were not

_{given explicitly.}

(iii)

One could

_give

another

_proof

of Theorem 2.4

_using

the

_theory

of normal bases. Recall first a result of Coleman

[8,

Lemma

A.1.2~

(extending

to a rank-one valuation a result of J.-P. Serre

[17,

lemme

1]

dealing

with the case where the valuation is

discrete)

which

_essentially

_says the

following:

a

family

lei I i E I }

of elements of

Mo

is a normal basis of M if and

only

if

there exists t E K with

_{It I}

1 such that the classes of the

_e~’s

form a basis

of the

_{V/tV-module Mo/tMo.}

In Theorem

_2.4,

M =

c(P, K)

and

Mo

=

C(2) V ).

It is then easy to

see that the

_{polynomials fn satisfy}

Coleman’s condition: it follows from

Proposition

1.3 that

C(E, V )

=

Int(E, V)

+ tC(E, V ),

thus,

the classes

of the

_{f n’s}

_generate

C (.9, V )/tC(E, V );

on the other

hand,

the

equalities

= 6k,n

for 0 k n

_{easily imply}

that these classes are

linearly

independent.

Finally,

we

_generalize

[3,

Theorem

2]

and

[18,

Theorem

3.3]:

Theorem 2.7. Let K be the

_quotient

_{field of}

a rank-one valuation domain

V,

and E be an

infinite

subset

of K

such

that.9

is

_compact.

Then, f or

_every

set

_{o f}

_generators

_{CPn}nEN

_{o f}

the V-module

_{Int (E,}

_V)

and _every continuous there exists a _sequence such

that 0 =

with

_{lim bn =}

0.

Moreover, if

the

_{Wn’s f orm}

a basis

of

the

- _n-&#x3E;oo

has:

In other

_words,

if

E

is

_compact,

then _every basis of the V-module

Int(E,

V)

is a normal basis of the ultrametric Banach _space

K).

Proo f.

From the series

_{expansion 0 =}

_[Theorem

_2.4],

we derive

another

_{one: 0 =}

where

_{b k}

=

E- n= 0

anbk,n,

with

_bk,n

E V

[Lemma

2.3].

Note

_that,

for such a _sequence we have the

inequality

Finally,

we assume that the

_pn’s

form a basis of

Int(E, V),

and _prove that

the

_expansion

_E

_bnpn

is

_unique.

Consider two series

_expansion

_bnwn

and

_bnwn

of the same

function 0

and _assume,

by

_way of

contradiction,

that

_{bko -=I}

for some

ko.

Let t E V be such that

v(t)

&#x3E;

_{v(bko -}

_bko).

As and -~ _+oo, there is an

_integer

n such that

v(bk -

b~ ) &#x3E;

v (t)

for

k &#x3E; r~

_(note

that n &#x3E;

_{ko ) .}

Consider the

_{polynomial1/J =}

As we can also

write

it follows

that 1/J belongs

to

_{tInt(E, V).}

Since the

_{decomposition of 0 along}

the basis formed

_by

the

Wn’s

is

_unique,

we have in

_particular

v (t) .

We thus reach a

contradiction.

The first

_part

of the

_proof

shows that &#x3E; It follows from Lemma 2.5 that we have an

equality.

0

3. The

_compactness

of

2?

is _necessary

We have

_already

noted

_that,

if f

is

_compact,

then E is a fractional subset

of V. On the other

_hand,

if E is not a fractional

_subset,

then

Int(E, V)

= V

[4,

Proposition

1.1.9],

hence non-constant continuous functions cannot be

approximated by

polynomials

in

_{Int (E, V) .}

We shall thus _suppose that E is a fractional

_subset,

and

_replacing

E

_by

dE

(which

is

homeomorphic

to

it),

that it is even a subset of V

(see

also

[4,

Remark

1.1.11]).

We first

_give

a characterization of the subsets E such that

E

is

_compact,

similar to

_[5,

Lemma

_{4.4] (see}

also

_[4,

_Proposition

_111.1.2]).

For

_this,

recall that the non-zero ideals of V are either of the form

Ia

=

{x

_E V

v (x) &#x3E;

o:},

or

Ia

=

{x

_E V

I v(x)

_&#x3E;

a~,

for _a real and

positive.

Correspondingly,

the

cosets modulo

_Ia

are closed balls of the

_type

Ba (a),

and the cosets modulo

la

are the

_{corresponding}

open balls. In any case, since K is an ultrametric

space, each such coset is a

_clopen

set.

Lemma 3.1. Let V be a rank-one valuation domain and E be a subset

of

V. The

_following

assertions are

equivalent.

2. For each non-zero ideal I

of

V,

E meets

only finitely

_many cosets

of

V

modulo I.

3. For each

_positive

_integer

_n, E meets

_{onl y finitel y}

_many cosets

_of

V modulo the ideal

_In

=

{x

_E V

( v(x) &#x3E; nl.

Proof.

1 ~ 2. The condition is

_obviously

necessary: if E meets

infinitely

many cosets modulo an ideal

_I,

then E is covered

_{by infinitely}

_many

disjoint

clopen subsets,

and its

completion P

is then also covered

_by

the

_disjoint

union of the

_completions

of these

_clopen

subsets.

2 ~ 3. This is obvious.

3 - 1.

_By

_hypothesis,

for each _n, # meets

_{only finitely}

many cosets of

V

modulo

In.

Let be a _{sequence in}

E.

Infinitely

_many of its

_terms,

forming

a subset

_Xl

of

k,

are in the same class modulo

7i.

_Infinitely

_many

terms of

_Xl,

_forming

a subset

_X2

of

_Xl,

are in the same class modulo

f2-And so on. We thus define a

decreasing

_sequence

(Xn)

of

subsets,

the

elements of

_Xn

_being

in the same class modulo

In.

Let x~n be the first term

of the _sequence which is in

_Xn.

The

_subsequence

is then a

Cauchy

sequence in

2

and it converges. 0

We shall write that is infinite

_(resp.

_finite)

if E meets

_infinitely

_many

(resp.

only

finitely many)

cosets modulo I.

Lemma 3.2. Let V be a rank-one valuations

domain,

with

quotient

field

K,

and let E be a

_fractional

subset

of

V.

If

Int(E, V)

is dense in

C(.9,

V ),

then the

_{completion E of E}

is

_compact.

Proo f .

Assume that E is a subset of V

and, by

_way of

contradiction,

that

2

is not

_compact,

that

_is,

_E/Ia

is

_infinite,

for some a &#x3E; 0. The

_principle

of the

_proof

is the

_following:

We consider a _sequence

fxnl

of elements of E in distinct classes modulo some

_Ia

(or

some It is thus

_possible

to define a

function 0

E

C(E, V )

such

_{that, alternatively,}

for n even and n

odd,

we have

§(zn) =

0 and

O(xn)

= 1-

Suppose

that

Int(E,

V)

is dense in

V):

in

particular,

there

exists

_f

E

_K[X]

such

_{that, alternatively,}

_{v ( f (xn))}

&#x3E; 1 and

_{v ( f(zn))}

= 0.

We reach a contradiction

by

choosing

the _sequence in a such a _way

that,

for each

_f

E

_K~X~,

the _{sequence converges. In}

_fact,

as we can write

f

= _a

gk) in

an

_algebraic

extension L of

_K,

it is

enough

to make sure

that,

for ~

E

_L,

the sequence

~) }

converges

(in

1~,

extending

the valuation v to a rank-one valuation of

L).

If is

_{infinite, then,}

a

fortiori,

and

EII’O

are infinite for

(3

&#x3E; a.

If the valuation is discrete

_(with

_{value group}

_Z),

_{then 7}

is an

integer

and

necessarily

E/I,y

is finite while is infinite. We then consider three

cases:

1)

E / I’Y

is

_finite,

and is infinite: there is a _sequence in E such

that all the are in the same class modulo but in distinct classes

modulo For _m, we then have

v (xn -

_{Let C}

be an element

of an

_algebraic

extension L of K. Therefore

-

either _y for some then

g)

=

g)

for all _n; -

or _{&#x3E; ’1} for some _no, then for

n ~

_no;

-

or _’1 for all n.

At any

rate,

the _sequence

_~)}

is

_eventually

constant.

2)

is finite. In this case, the valuation is not discrete and

E/h

is

infinite for each

_#

&#x3E; _7. Let

_(Qn)

be a

_{strictly decreasing}

_{sequence in} 1~

converging

to _’1.

_Among

the

_(finitely

_many)

classes that E meets modulo there is one

_which,

for each _{n, meets}

_infinitely

_many classes

_moduloan

By

induction,

we can thus build a _sequence

fxnl

such that all its terms are

in the same class modulo

I§

but _xn is not in the class of _{x1, X2,...} nor _xn-1

modulo For m &#x3E; _n, we thus

_have,

In

_particular,

all the

_xn’s

are in distinct classes modulo Moreover

-

either _~ for some _no, then

v (xn -

C)

= for all _n; -

or

_{ç) ~ ’1}

for all n and

_~)

&#x3E;

_Ono

for some _no; for n _{&#x3E; no} we then have

v (xn -

C)

=

xno ),

and

hence y :5

v (xn -

)

(3n;

-

or y :5

for all n.

Hence,

the _{sequence converges}

_{to q}

or is

_eventually

constant.

3)

is infinite. In this case

again

the valuation is not discrete and is finite for

_{3

q.

Here,

we let be a

_{strictly increasing}

_sequence in R

converging

to _~y. Let X be an infinite subset of

E,

the elements of which are in distinct classes modulo

_Infinitely

_many elements of

_X,

_forming

a

subset

_Xl

of X are in the same class modulo

_Ipl.

Infinitely

_many elements

of

_Xl,

_forming

a subset

X2

of

Xl,

are in the same class modulo

IP2.

And so on: we define a

_decreasing

_sequence

~Xn}

of

_subsets,

the elements of

Xn

being

in the same class modulo For each _n,

_choosing

_zn in

Xn,

we thus

build a _sequence such

that,

for m &#x3E; n, we

v(xm -

zn)

_7.

Therefore

-

either

_{8n,, for}

some _no, then =

~)

for

n &#x3E; _no;

-

or

_{~) &#x3E;}

_#n

for all n and

_{v(xno -}

_{~) ~ 7}

for some _no; for no

we then have

v (xn - ç)

= thus

7; -y for all n.

Hence,

the _{sequence converges}

_{to 7}

or is

_eventually

constant.

We conclude with the

_following.

Theorem 3.3. Let V be a rank-one valuations

domain,

with

quotient

field

K,

and let E be a

fractional

subset

of

V. The

following

assertions are

equiv-alent.

1. The

_completion

E

_of

E is

_compact.

2.

_K[X]

is dense in

c(.9, K),

_for

the

_uniform

_convergence

_topology.

3.

_{Int(E, V)}

is dense in

C(.9, V ),

_for

the

_uniform

convergence

topology.

4.

_Every

continuous

E G (E, K)

can be

expand ed

in tie

form

cngn,

with gn E

Int(E, V)

and

lim cn =

0.

n=

n--oo

5.

_Every

continuous E

C(E,

K)

is bounded.

Proof.

Replacing

E

_{by dE,}

we assume that E is a subset of the

_ring

V. That

1

_implies

_2,

3 and 4 was

proved

in the

previous

sections

[Proposition

1.3 &#x26;

Proposition

2.4].

That 4

_implies

5

_follows,

for

_instance,

from Lemma 2.5. And that 2

_implies

5 from the fact that the values of a

polynomial

are

clearly

bounded on a fractional subset.

Conversely,

5

_implies

1.

_Indeed,

assume

by

_way of

_{contradiction,}

that

2?

is not

_compact:

then.9

can be covered

by infinitely

_many

disjoint clopen

subsets,

and

_hence,

it is easy to define a continuous

(in

fact, locally

con-stant)

function which is not bounded.

Finally,

that 3

_implies

1 is the

_previous

lemma. 0 Remarks 3.4.

_{1) Analogously}

to

_[3,

Theorem

_3],

we derive an _easy

con-sequence of Theorem 2.7

(this

is also a _consequence of

[17,

_Prop.

3]):

Assume K =

K

and E =

E

is

_compact,

and let be a basis of

Int (E, V ) .

Then,

a K-valued measure on

_E,

that

_is,

a continuous K-linear

map p :

C(2?,~) -~

K,

is

uniquely

determined

by

the sequence of

its values on the basis

(that

is,

_pn =

if 0 =

E’o

_{anpn ,}

then

_{J.t( 4» =}

_E’o

_{Conversely, given}

a _sequence of

_K,

we claim

there is some _{measure p} such that _pn = if and

only

if this _sequence

is bounded. It remains to show that the boundedness is _necessary:

Assume that for a measure _p, the _sequence is not bounded. Let

t E V be such that

_v(t)

&#x3E; 0. There is an

increasing

_sequence of

positive integers

such that

_{-2nv (t);}

but

_then,

the value

_{of p}

cannot be defined on the continuous

function 4&#x3E; =

2)

We considered

_only

the case where E is

infinite,

although k

is

_compact

when E is finite. In

_fact,

it follows from

_{Lagrange interpolation}

that each function from a finite set to

K

is continuous and

_{"polynomial" :}

if E =

jai,

... , if Pj = for

1 j

r, and

if 0

E

C(E,

K),

then

ai-ai ,

4&#x3E; =

However,

this

_{representation}

is not

_unique:

for instance

the zero function _may be

represented by

the

polynomial f

for

ideal that there is no

_regular

basis of the V-module

Int (E,

V) (similar

to the basis

_{given by}

a

_v-ordering

of an infinite

subset).

3)

Let D be a

one-dimensional, local,

Noetherian domain with maximal

ideal n, such that the

completion

D of D in the n-adic

topology

is an

integral

domain. It is known

_that,

for a subset E of the

_quotient

field of

_D,

such that the

_topological

closure

E

of E is

_compact,

_{Int (E, D)}

is dense in the

_ring

_C(E,

D)

of continuous functions

from E

into

D [5,

_Proposition

_4.3].

Conversely again,

the

_compactness

of E

_{is necessary.}

_Indeed,

assume

that.9

is not

_compact.

Since D is an

integral domain,

the

integral

closure V of D is a discrete valuation domain with maximal ideal _m, and the m-adic

topology

induces the n-adic

_topology

on D.

Consequently, 2

is not

_compact

for the m-adic

_topology.

If E is not a fractional subset of

V,

then

Int(E,

D)

g

Int(E, V) =

V,

and

_hence,

_{Int(E, D)}

cannot be dense in

~{E, D).

If E is a fractional subset of

V,

it follows from the

proof

of Lemma 3.2 that some continuous

_{function §}

E

C(E, V ),

_{taking only}

the values 0 and

_1,

cannot be

_approximated

modulo

m

_by

a

polynomial f

E

_Int(E,

_V).

Since

its values are in

D,

then 0

belongs

to

C(E, D),

and a

fortiori,

it cannot be

approximated

modulo

n

_by

a

_polynomial

_f

E

_{Int{E, D).}

Acknowtedgments.

The authors are

_grateful

to the referee for

_interesting

references and comments with

_respect

to normal bases.

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Universit6 d’Aix-Marseille III CNRS UMR 6632 E-mail: _{paul-jean.cahenCuniv.u-3mrs.fr} Jean-Luc CHABERT Universite de Picardie CNRS UMR 6140 E-mail: _{jean-luc.chabertQu-picardie.fr}