Polynomial mappings defined by forms with a common factor

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

F. H

ALTER

-K

OCH

W

ŁADYSŁAW

N

ARKIEWICZ

Polynomial mappings defined by forms with

a common factor

Journal de Théorie des Nombres de Bordeaux, tome

4, n

o

_{2 (1992),}

p. 187-198

<http://www.numdam.org/item?id=JTNB_1992__4_2_187_0>

© Université Bordeaux 1, 1992, tous droits réservés.

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

Polynomial

mappings

defined

by

forms with

a common

factor

par F. HALTER-KOCH AND W. NARKIEWICZ

1. Introduction and Preliminaries

For a field

_Ii,

we denote

_{by K}

a fixed

algebraic

closure of K. For a

_point

x =

(x1, ... , xn~

_E

kn,

_we set

A subset V C

K’~

is called

_{K-homogeneous}

if there exist

_homogeneous

polynomials Hl,...,

Hr

E

_{~~~X1, ... , Xn~}

such that

For m , n e N and

_polynomials

E

_{.~~ ~X 1, ... , X ~ ~,}

we define

the

_{polynomial mapping}

and we set

We consider the

_following

finiteness

_property,

which a field Il _may have

or not:

An

_{algebraically}

closed or real closed field

clearly

never has the

_property

(PO).

As the main result of _{this paper,} we shall prove that _every

_finitely

generated

field has the

_property

_{(PO) (Theorem 3).}

This is well known if

Fo = 1

and

_additionally

either

_{global field,}

cf.

_{[5], [6].}

In

_[1]

we also dealt with the case

_{Fo = 1;}

there we considered a

_property

which is somewhat weaker than

_(PO)

and

_proved

it for a

larger

class of

fields. We also discussed the various

_hypotheses

stated in

_(PO),

we outlined

the

_history

of the

_problem,

and we _gave several further references.

The case 1 was _up to now

only

considered in a rather

special

_case,

cf.

_[7].

The

_following

_example

_(due

to the

_referee)

shows that the

_inequality

in

₍₂₎

cannot be weakened in a trivial _way: let K be an infinite

_field,

m = n =

_3,

V =

k3,

F =

(hence

Fo

=

X3),

G

-(XI,X2,X3)

and X =

_I

x, z E

K" ~

C

_V;

then

_{(1), (3)}

and

₍₅₎

are

_obviously

_satisfied,

and

₍₄₎

holds since

_(x,

_y,

_z)

=

F(x-1 z,

x-lz,

x3z-2 );

however,

deg~(F)

=

deg*(G)

₊

2deg(Fo),

and X is infinite.

The main tools in

_proving

that a field has the

_property

_(PO)

are the

theory

of

_height

functions

_(to

be

_explained

_{in §}

_2),

the

_theory

of

_places

of

_algebraic

function fields

_(§

₃₎

and the

_following

set-theoretical

_Lemma,

already proved

in

_[1].

MAIN LEMMA. Let

_{X, 3)}

be sets and

_F,

_{G : ~ -&#x3E; ~}

_mappings

such that

G(~).

Let

_{J : X --t}

R be a

_mapping

such that C R is

_discrete,

and let C &#x3E; 0 be a real constant with the

_property

that _{x , y}

_{E 3C,}

_{f (x) &#x3E;}

C

and

_F(x)

=

G(y)

implies

f (y)

_&#x3E;

then D

_G(Xc).

If moreover

3lc

is finite and G is

_injective,

then

X=Xc.

2.

_Heights

In this

_section,

we collate the _necessary facts from the

theory

of

height

functions,

following

[8].

Let I( be a

field, equipped

with a

family

of absolute values

satisfying

the

_product

formula. We

_regard

_M¡(

as a set of

_places

such

that,

for _{every v E} is an absolute value

_defining

_v; we denote

by

Iiv

the

_completion

of K at v.

For a finite extension field L of let

_ML

be the set of all

_places

of L

extending

those of

_M¡(.

If v E

ML

and w

_v,

then

where

_Lw

is a finite-dimensional local

lev-algebra

with residue field

L.

We define an absolute

value )’ w

of L

_by

Then

_{(I -}

satisfies the

_product

formula.

The

_family

M =

_gives

rise to _a

height

function H :

f,7n --,

R,

defined for _{every n E}

_N,

as follows: for x =

(xl, ... ,

_E

l’(n,

let L be a

finite extension of K such that x E

_Ln,

and set

Thanks to the normalization this definition does not

_depend

on the choice of the field L. We call H =

HM

the

height function

associated with

M.

For a monic

polynomial

of

_degree

_{d E fi1,}

we set

LEMMA 1. Let K be a

field, equipped

with a

family

of absolute values

satisfying

the

_product

_formula,

and let H be the associated

_height

function. If n

_{e N,}

x E

.Kn

and z E then we have

assertion is

_equivalent

with

and hence also with

Since

_obviously

and

_(by

the

_product

_formula)

it is sufficient to observe that 0 a 1 and b &#x3E; 0

_implies

L~

The next Lemma

_gives

information about the behaviour of

_heights

under

polynomial mappings.

,

LEMMA 2. Let il be a

_{field, equipped}

with a

family

of absolute values

satisfying

the

_product

_formula,

and let H be the associated

_height

function. Let F =

(Fl , ... , Fm ) ;

Î(n

--~

¡em

be _a

_{polynomial mapping,}

_defined

_by

homogeneous

polynomials Fl,. - .,

F~

E

_{~~~X1, ... ,}

_Xn~,

d* =

_deg*(F)

and

d* =

i)

There exists a constant

C1

E such

_that,

for all x E

7~~~

ii)

Let V C be a

I( -homogeneous

set such that

_{F1, ... , F~.,.L}

do not

have a common non-trivial zero in V. Then there exists a constant

C2

E such

_that,

for all x E

_V,

Proof.

i)

follows almost

_immediately

from the definition of the

_height

_function,

cf.

_{[8, 2.3].}

ii)

is

_proved

in

_[5,

_Proposition

_2].

0

PROPOSITION 1. be a

field, equipped

with a

family

of absolute

values

_satisfying

the

_product

_formula,

and let H be the associated

_height

function.

For m, n

(Fl,... 1 Fm)

7 G -

(Gi Gm) : k- , Î(m

be

_{polynomial mappings}

defined

_{by homogeneous polynomials}

_{Fj ,}

_Gj

E

Let Fo

E be a common factor of

Fl,...,

say

FOF§ where Fj

E

_{.~[Xl, ... , Xn~,}

and suppose that

deg*(F)

&#x3E;

deg*(G)

+

_{2 deg(Fo).}

Let V C

Pi’

be a

_{K-homogeneous}

set such that

Fi, ... , F,~

do not have a common non-trivial zero in V.

i)

There

exists a constant

Co

E

_R&#x3E;o

such that _{x , y} E

_V,

_{H(x) &#x3E;}

Co

and

_F(x)

=

G(y)

implies

H(y)

_&#x3E;

H(x).

ii)

Let

Co

be a constant

_satisfying

_{i) ,}

~ C V a subset such that

G(X),

GIX

is

_injective,

and

Xo =

f x

I

H(x)

Co}

is finite.

Then X is finite,.

for all x E

_By

Lemma

_2,

there exist constants

_{Cl ,}

_C2

E

_R&#x3E;o

such

that _{x , y} E ~ and

_F(x)

=

G(y)

implies

and hence

_H(y)

&#x3E;

_H(x),

_provided

that

_H(x)

is

_{large enough.}

ii)

follows

_immediately

from

_i)

and the Main

_{Lemma, applied}

for

_{F ,}

G :

~E-~~t’n

and

_f

=H:~-~I~. C7

THEOREM 1. Let il be a

field, equipped

with a

family

of absolute values

satisfying

the

_{product formula,}

and let H be the associated

_height

function.

Suppose

that for _every C E

_R&#x3E;o

and n E

_fiT,

the set

is finite. Then Ii has the

_property

_(PO).

In

_particular,

every

global

field

has the

_property

_(PO).

Proof.

The first assertion follows

_immediately

from

_Proposition

1. If Il is

a

_global

field

(i.

_e., either an

_algebraic

number field or an

_algebraic

function field in one variable over a finite

field),

then we use the finiteness results

from

_{[8, 2.5].}

0

We close this section with two further Lemmas

_{concerning heights}

to be used

_{in §}

3. For

_simplicity,

we restrict ourselves to the case of

non-archimedian absolute values.

LEMMA 3. Let I~ be a

field,

equipped

with a

family

of non-archimedeall

absolute values

_satisfying

the

_product

_formula,

and let H be the associated

height

function.

i)

For monic

_{polynomials f , g}

E we have

_{H( fg) H( f )H(g~.}

Proof.

i)

is

_obvious;

we even have

_equality,

cf.

_[2,

ch.

_3,

_Prop.

_2.4].

ii)

Set x =

(x1, ... , xn ),

_y =

( yi , ... , yn ),

and let L be a finite extension

of Ii such that _{x, y} E Ln. Then we have

LEMMA 4. be a

field, equipped

with a

family

of non-archimedean

absolute values

_satisfying

the

_{product formula,}

and let H be the associated

height

function.

Let

_{d , n E h1,}

and let x =

(x1, ... , xn)

_E be such that

dI~(x)

d.

For i E

_{~l, ... , n~,}

let fi

E be the minimal

polynomial

of xi,

and

f = fi

-...-fn

E Then we have

where

E Ie

are the

_conjugates

of xi

(counted

with

multiplicity

in the

inseparable

case)

and

_{di ~}

d. Since

_{H(X -}

=

H(xi ),

_we

obtain, using

Lemma

_3,

3.

_Algebraic

function fields

We start with some

preliminaries

on

height

functions and

places

of

alge-braic function

_fields;

for the

_general

_theory

of

_places

cf.

_[3,

ch.

_I].

Let

_Kl

=

Ko(t)

be _a rational function field over an

_arbitrary

field

_{Iio ,}

and let P be the set of all monic irreducible

_polynomials

in For

p E P,

we define an absolute

value j’

jp

as follows: If

where c E

~~

_{vp 6 Z} and _vp = 0 for almost all then

Then

_{() ’}

is a

family

of non-archimedean absolute values

sat-isfying

the

_product

formula. The

_{corresponding height}

function will be denoted

_by

_Ha;

we call it the canonical

height function for

If x e

_1(1’

then x =

(al,8-I,...,an,B-I),

where _E

{3

is monic and _{an ,}

_,8)

=

1;

this

implies

and hence

where

Every

r E

ko

defines a

_place

_{~T : K,}

--~

ko

_by

_0,(h)

=

1~ (T ) .

A

place 0:

- is called _a

_I(o-

_{place if}

_OIK,

=

§r

for _{some 7 E}

-f!Co.

Obviously,

every

(T

E

Î(o)

extends to a

I(o-place

of

K(I.

For a

ICO-place

0

of

_I(l,

we denote

_by

_{R~ _}

C the local

_ring

associated with

0.

We extend the

_ring

_{homomorphism 0: R --}

to

_{homomorphisms}

of the

polynomial

rings 0 :

(by

acting

on the

PROPOSITION 2. Let

_Iio

be a field which is neither

_{algebraically}

closed nor real

_closed,

let

_{K, =}

be a rational function field over and

_Ha

the canonical

_height

function.

For _every there exists a

~~1

-~

Pio

with the

following

property:

and Z C a subset

_satisfying

then we have Z C

_R’,

and

_{~~,~ : .~ ~}

_injective.

Proof.

Let r E

I(o

be an element

_satisfying

and

_{let 0 :}

_Kl

2013

ko

U

_{oo}

a

place extending

0,.

We assert

_{that 0}

has the

required

property.

Let m e N and Z C

K’’n

be a subset such that

Ha(z)

+ + m M

for all z E .~.

_Putting

it is sufficient to _prove that ,~* C

_Rm,

and

_{~(x) ~}

0 for all

_{0 ~}

x E ~*.

For x E

_Z*,

we have

M2,

and

Ha(x)

M2

by

Lemma 3. For x =

(xl, ... , x."i ),

_we denote

by Ii

_E the minimal

polynomial

of _~2,

and we set

where _{ao, ... ,}

_~3

E is monic and

_gcd,

Lemma 4

_implies

integral

over

_Rp,

and since

Ro

is

integrally

closed,

we obtain x E

If

_{x #}

_0,

then we 0 for

some j

m~

and 0 for some i E

_{(0, ... , d -1 ~.}

Let I E

_{f 0,..., d -1 }}

be minimal such tha,t

0;

then we obtain

Dividing

by

_x~

and

_multiplying

_{with 0 yields}

an

equation

Since

_{~(al ) -}

=

_0,

_we obtain 0 and hence

~(x) ~

0. D

THEOREM 2. Let

₁₍₀

be a

_fields,

and _suppose that _every finite extension field o~’

_I(o

has the

_property

_(PO).

Then _every

_{finitely generated}

extension

field has the

_property

_(PO).

Proof.

It is sufficient _{to prove} that _every

_{finitely generated}

extension field of transcendence

_degree

1 over the

_property

_(PO);

the

_general

case follows

_by

an obvious induction on the transcendence

degree.

Let =

Ilo(t)

be _a rational function field _over and let 7~ be _a finite

extension of We shall _prove that K has the

_property

_(PO);

_obviously,

we

have k

=

kl.

Let

Ho

be the canonical

height

function associated with

~~o (t).

,

Let F =

_{(F1, ... , Fm ),}

G =

_{(G1, ... , Gm ) :}

Pi,’

--~

lem

be

_polynomial

mappings

defined

_by

_homogeneous

_polynomials

_{Fj, Gj}

E

_{7~[Xi,...,~].}

Let

Fo

E

_{.~~~X1, ... , X~~}

be a common factor of _say

_{Fj =}

where

_F~

E

_{~i ~X1, ... , Xn~,}

and _suppose that

_deg*(F)

&#x3E;

_deg*(G)

+

2 deg(Fo).

Let V C be a

_{]( - homogeneous}

set such that

_{F1, ... , F£}

do not have a common non-trivial zero in

V,

and

let X

_{C V} be a subset such

that and

_GIX

is

_{injective. By Proposition}

_1,

there exists a

constant

_Co

E

_R&#x3E;o

such that _{x , y}

_{Ha(x) &#x3E;}

_Co

and

_F(x)

=

G(y)

the Main Lemma

_yields

_{F(%o) D G(Xo)?}

and

_according

to

_Proposition

1 it is sufficient to _prove that

Xo

is finite.

_By

Lemma

_2,

there exists a constant

Cl E R&#x3E;o

such that

for all x ~

Xo.

Let

_{H1, ... ,}

_Hr

E

_{]([XI,... ,}

_Xn]

be

_homogeneous

_polynomials

such that

n

I .~l~x~ - ... - Hrtx~ - 0~.

Since

_{Fl’,..., F.,}

_HI,...,Hr

do not have a common non-trivial zero in

Pi n

it follows from Hilbert’s

Null-stellensatz

_[9,

_§130]

that there exist

_polynomials

_Xn

such that -

-for some _{exponent q} E N and all v E

_{{I,... , n}.}

Let D be the

_(finite)

set of all coefficients of the

_{polynomials Fo ,}

_Fj,

Hj , Fjv

and Since has the

_property

_(PO),

it is neither

alge-braically

closed nor real

_closed,

and

_Proposition

2

_applies:

there exists a

.~1’ -~

U such that D C

_RO,

Xo

C

_Rn,

_{ol D}

U

_~0~

is

injective

and is

_injective.

There exist elements _{yl , ... , yk E}

_Ro

such that

_{ICO (t,}

_YI,

_Yk)

and

_consequently

_{~(~i )}

= L _U

{oo},

where L =

1(0 ( Ø( t),

0(yl),.. - ,

L is a finite extension of

Ilo,

thus it has the

_property

(PO).

We use the

place 0

to shift the whole situation from K to L. We have

i

= the

polynomials Fo ,

_{F) ,}

Hj , Fjv

and

_H~v

have coefficients in and

consequently

their

_0-images

have coefficients in L.

We consider the

_polynomial

_mappings

_{~(F) - (~(F1 ), ... , ~(F~ ))}

and ;

by

construction one has:

then

_Vo

C

_kg

is a

K-homogeneous

_set,

and since

the

_polynomials

_{~(Fi ), ... , ~(F;L )}

do not have a common non-trivial zero

Obviously,

we have contend that

and we

are both

injective. Indeed,

if _{x, y} E then

_{4&#x3E;(G(x)) =}

_§(G(y))

and hence both

_injective.

Since L has the

_property

_(PO),

we conclude that is

_finite,

and hence

Xo

is also finite. 0

THEOREM 3.

_Every

_finitely

_generated

field has the

_property

_(PO).

Proof.

By

Theorem

_2,

it suffices to _prove that _every finite extension of a

prime

field has the

_property

_(PO).

For finite

_fields,

this is

_obvious;

for

algebraic

number

_fields,

it follows from Theorem 1. 0

REFERENCES

[1]

F. _Halter-Koch, W. Narkiewicz, Finiteness _{properties of polynomial mappings,} to

appear.

[2]

S. _Lang, Fundamentals _{of Diophantine}

_Geometry,

_Springer 1983.

[3]

S. _Lang, Introduction to _{algebraic geometry,} Interscience Publ. 1958.

[4]

D. J. Lewis, Invariant sets _{of morphisms} in _projective and _affine number spaces, J. Algebra 20

_(1972),

419-434.

[5]

P. _Liardet, Sur les _{transformations polynomiales} et _{rationnelles,} Sém. Th. Nomb. Bordeaux exp. n° 29, 1971-1972.

[6]

P. _Liardet, Sur une conjecture de W. _Narkiewicz, C. R. Acad. Sc. Paris 274

_(1972),

1836-1838.

[7]

W. _Narkiewicz, On _{transformations by polynomials} in two _{variables, II,} Coll. Math. 13

_(1964),

101-106.

[8]

J.-P. _Serre, Lectures on the Mordell-Weil-Theorem, Aspects of Mathematics, Braun

schweig 1989.

[9]

B. L. van der _{Waerden, Algebra,} 2. _{Teil, Springer} 1967.

Franz Halter-Koch Institut fur Mathematik

Karl- Franz ens- Universi tåt Hehlridhstrafle

_36/IV

A-8010 Graz,

4sterreich

Wladyslaw Narkiewicz

Instytut Matematyczny Uniwersytet Wroclawski Plac Grunwaldzki _2/4