Linear forms in the logarithms of three positive rational numbers

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

URTIS

D. B

ENNETT

J

OSEF

B

LASS

A. M. W. G

LASS

D

AVID

B. M

ERONK

R

AY

P. S

TEINER

Linear forms in the logarithms of three positive

rational numbers

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{1 (1997),}

p. 97-136

<http://www.numdam.org/item?id=JTNB_1997__9_1_97_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

Linear forms in the

_logarithms

of three

_positive

rational numbers

par CURTIS D.

BENNETT,

JOSEF

BLASS,

A.M.W

GLASS,

DAVID B. MERONK et RAY P. STEINER*

To Hassoon S. Al-Amiri upon his retirement

RÉSUMÉ. Dans cet _article, nous donnons une minoration de la _dépendance

linéaire de trois nombres rationnels positifs valable sous certaines conditions faibles d’indépendance linéaire des coefficients des formes linéaires. Soit A = _{b2 log 03B12 -} b1 log 03B11 - b3log 03B13 ~ 0 avec b1, b2, b3 des entiers positifs et 03B11, 03B12, 03B13 des rationnels multiplicativement indépendants supérieurs à 1.

Soit 03B1j =

_{03B1j1/03B1j2}

où 03B1j1, 03B1j2 sont des entiers _{positifs premiers} entre eux

(j = 1, 2, 3). Soit _{aj ~}

max{03B1j1,e}

_{et supposons que} pgcd(b1, b2, b3) = 1.

Soit

(b2 log03B13 + b3 log03B12)

et _{supposons que B ~ max{10, log}

_b’}.

Nous démontrons _que, soit

_{b1,

_b2,

_b3}

est _{(c4, B)-linéairement dépendant} sur Z _{(relativement} à _(a1, a2, a3)), ou bien

3

où c4 et C = _{c1c2 log 03C1+03B4 sont donnés dans les tables de la Section} 6.

Ici nous dirons _{que b1, b2, b3} sont _{(c, B)-lineairement dépendants} sur Z si

d1b1 + d2b2 + d3b3 = 0 _pour certains d1, d2, d3 E Z non tous nuls tels que ou bien _(i) 0

|d2| ~

cB _{loga2 min{log a1, log a3},}

_|d1|,

_{|d3| ~} cB _{log a1 log a3,} ou bien _{(ii) d2} = 0 _{et |d1| ~ cB log a1 log} a2 et

|d3| ~

cB log a2 log a3.

Nous obtenons en _particulier c4 9146 and C 422321 pour tout B &#x3E;

10, et si _{B ~} 100 nous avons _{c4 ~} 5572 et _{C ~} 260690. Des informations

plus précises sont données dans les tables de la Section 6.

Nous démontrons ce résultat en modifiant les méthodes de P. _Philippon,

M. _Waldschmidt, G. Wüstholz, et al. En _particulier, par un _argument

combinatoire, nous _{prouvons que} soit une certaine variété _algébrique est de dimension _nulle, ou bien

_{b1,b2,b3}

sont linéairement _dépendants sur Z, Manuscrit _reçu le 26 octobre 1995

*

The authors are _grateful to NSF and MSRI for partial funding. We wish to thank Alan Baker FRS for the invitation to _participate in the Transcendental Number _Theory

program at MSRI last _year, and the staff there for their warm _hospitality. This paper grew out of several discussions there.

avec de _petits coefficients de dépendance. Cela nous _permet d’améliorer le

Lemme de zéros de Philippon, nous conduisant au fait _que le déterminant

d’interpolation reste non nul sous des conditions plus faibles.

ABSTRACT. In this paper we _prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear _independence conditions on the coefficients of the linear forms. Let A = b2 log _03B12 -b1 log 03B11 - b3 log03B13 ~ 0 with b1, b2, b3 positive integers and 03B11, 03B12, 03B13

positive multiplicatively independent rational numbers _greater than 1. Let 03B1j =

_{03B1j1/03B1j2}

with 03B1j1, 03B1j2 _{coprime positive integers (j} = 1, 2, 3). Let

aj ~

max {03B1j1, e}

and assume that _{gcd(b1, b2,} b3) =1. Let

and assume

max{10, log b’}.

We prove that either

{b1,b2,b3}

is

(c4, B)-linearly dependent over Z _{(with respect} to _(a1, a2, a3)) or

3

where c4 and C = _{c1 c2 log 03C1+03B4} are _given in the tables of Section 6. Here

b1, b2, b3 are said to be _{(c, B)-linearly dependent} over Z if _d1b1 + d2b2 +

d3b3 = 0 for some _{d1, d2, d3} ~ Z not all 0 with either

(i) 0 _{|d2| ~ cB log a2}

_{min{loga1,loga3}, |d1|,}

_{|d3| ~}

_{cB loga1 loga3,} or

(ii) d2 = 0 and

_{|d1| ~}

_{cB log a1 log} a2 and

|d3| ~

cB log a2 log a3.

In _particular, we obtain c4 9146 and C 422,321 for all values of B ~ 10, and for _{B ~} 100 we have _{c4 ~} 5572 and C 260,690. More

complete information is _given in the tables in Section 6.

We prove this theorem by modifying the methods of P. Philippon, M.

Wald-schmidt, G. _Wüstholz, et al. In _{particular, using} a combinatorial

argu-ment, we _prove that either a certain _algebraic variety has dimension 0 or

{b1,b2,b3}

are _{linearly dependent} over Z where the dependence has small coefficients. This allows us to _{improve Philippon’s} zero _{estimate, leading}

to the _{interpolation} determinant _being non-zero under weaker conditions.

In the 1930’s Gel’fond and Schneider

_{independently}

studied linear forms

in the

_logarithms

of two

_algebraic

numbers in order to answer Hilbert’s

7th

_problem,

a

_special

case of which is: Is

2"

transcendental? The

_study

of linear forms in the

_logarithms

of more than two

_algebraic

numbers is far more

_complicated.

The

_{great pioneer}

work of Alan Baker came in the

1960’s. He

_proved

that any finite set of

_logarithms

of non-zero

_algebraic

numbers is

_{linearly dependent}

_{over Q}

_(the

field of

_algebraic

_numbers)

if and

_only

if it is

_{linearly dependent over Q}

_(the

field of rational

_numbers).

Moreover,

he showed that if

is not zero and

then

_IAI

can be

_explicitly

bounded _away from zero. He

improved

this in

[1]

(in

the case that

bo

= 0 and

bi, ... ,

bn

_are rational

integers)

_to:

for some

_{explicitly computed}

large

constant C. Here

Further,

h(a)

is the absolute

_{logarithmic height}

of a, and for any field

K,

K* denotes the

_{multiplicative}

_{group of} non-zero elements of K.

For the rest of the paper, we will assume that

_bo

= 0 and

bl, ... , bn

_are

rational

_integers.

From these

_results,

one can derive upper bounds on the size of the

so-lutions of _many

_{Diophantine equations}

_by

a function C

_dependent

_upon C

and the

particular

equation

(see

e.g.,

[26], [28]

and

[8]);

the smaller the value of

_C,

the smaller the

_potential

solution set

_{(though C(C)}

is

_typically

very

large).

For this

(and other)

reasons, many mathematicians have

spent

the better

_part

of the _{last 25 years in} an

_attempt

to decrease the value of C

and extend Baker’s work to other contexts

_(e.g.,

the

_theory

of linear forms

in

_{p-adic logarithms}

_{[24], [38], [39], [6],}

and

_elliptic

_logarithms

_{[25], [23],}

[27~).

By

1990 successive work

_{by Baker,}

Waldschmidt and

_others,

_~l],

_{[29], [3]}

had reduced C to

_{n 2n+l (24e 2)n220D n+2 .}

The next

_major

_{breakthrough,}

the use of

_algebraic

_groups and

_algebraic

_varieties,

was

developed

in the

1980’s.

_Impetus

for this work came from Nesterenko

[20],

who used

Brownawell and Masser

_[4]

_developed

this idea further and Masser

_[15]

and Masser and Wiistholz

_[16],

_{[17], [18]}

_replaced

the idea of derivations

_by

the

use of

_{ring homomorphisms.}

Wüstholz

[34]

developed

these ideas further

in our

_setting

and achieved

significant

results on zero

_{and"multiplicity"}

estimates that were later refined

by Philippon

[21]

and Denis

[5].

The

explicit application

to the

_theory

of linear forms in

_logarithms

was

_given

by

W3stholz

_[35],

_{[36], [37]}

and

_Philippon

and Waldschmidt

_[22].

_Finally,

Baker and Wüstholz

_[2]

and Waldschmidt

_[32]

combined the new ideas with

the

_previous

methods of Gel’fond &#x26; Baker and

_Schneider,

_respectively

_(for

bo

= 0 and all _are rational

integers).

Baker and Wiistholz obtained

C =

18(n +

1)!nn+1(32D)’~+Z log(2nD).

This is

currently

the best result known and is an

_improvement

on Waldschmidt’s

independent

1993 _paper

[32]

by

a factor of

approximately

Recently,

Michel Laurent

_{[11], [12], [13]}

_developed

a new

technique,

In-terpolation

Determinants;

this

_replaces

_(and

is somewhat

_analogous

_to)

the construction of an

_auxiliary

function. In the case of two

_logarithms

with

bo

=

0, Laurent,

Mignotte

and Nesterenko

[14]

obtain C 50 if b’ _&#x3E; 16 and C

_approaches

15.2 as b’

_approaches

_infinity,

where b’ =

log

+

ug

_(in

the

log a2 log al

case that _al and a2 are rational numbers

_greater

than

_1,

and all

logarithms

are

principal).

This was a

significant

improvement

on the

_previous

bound

of 270

_{given by Mignotte}

and Waldschmidt

_[19].

_{Unfortunately,}

in both papers

log b

is

replaced by

(log

b’)2.

In the case

bo

= 0 and _{n &#x3E;} 2

Wald-schmidt

_[33]

obtained _upper bounds on C

_by

this method.

Again log

b’ is

replaced by

(log

b’)2,

though

this can be circumvented at the _expense of a

larger

value of C

_[32].

As

_{already noted,}

to obtain solutions of certain

_{Diophantine equations,}

one needs smaller values of C. This motivates our

study

of linear forms

in the

_logarithms

of three

_algebraic

numbers.

_(All

our

_assumptions

will be motivated

_by

such

_{considerations.)}

As Gel’fond

_[7,

_page

_177]

stated in

_1952,

non-trivial effective estimates for linear forms in three or more

_logarithms

of

algebraic

numbers would lead to

_great

advances in the

_theory

of

_numbers,

in

_particular

the

_computation

of

_explicit

bounds for the solution set of certain

_{exponential Diophantine}

_equations.

_Indeed,

he showed how to

_apply

these ideas to the solution of Thue

_equations.

At that

_time,

the _necessary estimates were known for the case of two

_logarithms

but the transition to

three

_logarithms

_presented

considerable

_difficulty

and had not been carried out. Baker’s

_insight

provided

the

_original

method and solution for the case

of three

_{logarithms (and}

the more

_general

case

involving n logarithms).

The

purpose of this paper is to concentrate on the

_particular

case of linear forms

and

bo -

0. The treatment of the

_general

case

(n

&#x3E;

3,

logarithms

of

algebraic

numbers,

and

_algebraic

_{coefficients)}

_by

_{interpolation}

determinants is considered

_by

Lisa Elderbrock in her Ph.D. dissertation. For n =

3,

_an

extra

D5

is

_needed,

but the constant is reduced

_{by approximately}

D3.

Often,

bounding

solutions of

_{Diophantine equations}

involves

_bounding

the coefficients of the

_logarithms

in the

_resulting

linear

_forms;

see

[26]

for several

_examples.

If these coefficients

_satisfy

a linear

dependence

rela-tion over Z with "small"

coefficients,

then we

usually

obtain much

sharper

bounds on the size of solutions

(to

the

Diophantine

equation)

than those

provided

by

bounding

the linear form in

_logarithms

_away from 0.

Conse-quently,

we will

_always

assume that the coefficients of the linear form are

not

_{"strongly linearly dependent"}

over Z.

To

_simplify

matters, we assume that

fal,

_a2,

a3l

is a

_{multiplicatively}

independent

set of

_positive

rational numbers and that is a set

of

_{coprime integers.}

Let A =

bl log

_{al +}

b2 log

_{a2 +}

b3 log

a3 ~

0. If all

-

1, 2, 3)

are

_positive,

then A &#x3E;

_bl

_log

_al +

_b2

_log

a2.

Using

[14],

we can bound A "far" _away from 0 in this

_{(two logarithm)}

case.

Simi-larly

if all

_{bj log aj}

are

_negative,

can be bounded "far" _away from 0.

Furthermore,

since

_{bj log O!j =}

_-bj

we can assume that each of

al, cx2, a3

strictly

exceeds

1,

that

bl, b2, b3

are all

_positive

_integers

with

highest

common factor

_1,

and

A =

b2

log

_{a2 -}

b1log

_{al -}

b3

log

a3 ~

0;

moreover, we _may assume that

_{b2 log}

_{a2 &#x3E;}

max{bllog

_{aI, b310g}

_a3}.

Let ai , a2, a3 be real numbers

greater

than or

equal

_{to e;} without loss of

generality,

let a3. Let

and B &#x3E;

_{log b’.}

We say that

b3}

is

(c, B)-linearly

dependent

over Z

(with

respect

to

_{(~1,~2 a3)) if}

THEOREM A. Let A =

b2 log

_{a2 -}

b110g a1 -

b3 log a3

0 with

bl, b2, b3

positive

rational

_integers

and _{a1, a2, a3}

_positive

_{multiplicatively}

indepen-dent rational numbers

greater

than 1.

_{Let aj =}

_{with aj1, aj2}

coprime

positive integers

( j

=

1, 2, 3).

=

1, 2, 3)

and assume that =1..Let

and B &#x3E;

_{max{10, log b’}.}

THEN either

_(104,

_B)-linearly

de-pendent

over Z

(with

_respect

to

_(al,

_az,

_a3))

or

where C = 4.5 x

105 .

Theorem A follows at once from the more technical theorem:

THEOREM B. Let A =

b2log

a2-b1

log

log

a3 #

0 with

61,

b2, b3

POS-itive

_integers

and al, a2, a3

positive

muItiplicatively independent

rational

numbers

_greater

than 1. _{Let aj} = with ajl, aj2

coprime positive

integers

(j

=

1, 2, 3).

=

1, 2, 3),

and _assume that

=1. Let b’ =

( b2

₊

-)(-

_{+ assume} that

log al log a2 log a3 log a2

B &#x3E;

_{maxBo, log b’}.}

Either is

_{(c4, B)- Iinearly}

_dependent

over

Z

_(with

_respect

to

_{(aI, a2, a3))}

or

where c4 &#x26; C = _{Cl C2}

109 P + 6

_are

_{given by}

Table 2 in Section 6. In

_comparison,

Baker and Wustholz

_[2]

obtain

’"

with C 1.5 x

1012 ,

where b = =

1,2,3}.

Part of the technical difhculties _{to prove} this theorem occur in Section 3.

They

arise when we

_try

to take full

_advantage

of the weak linear

depen-dence. If we remove such

_assumptions,

the easier

_portion

of Section 3

_yields

a constant C

_(in

both Theorems A and

_B)

that is 8 times the value stated.

Clearly,

the result of Baker and W3stholz is much

_tighter

than our the-orems in terms of the

_parameters;

so it is much nicer both

_{theoretically}

and for

_large

values of

_log

b and

_{log b’ (since}

we have an extra

_log

b’ in our

result). (Thus

our table is

only

for

log b’

106 . )

However,

in many cases our result is more useful for

applications,

for

example

when

solving

certain

Diophantine

equations.

Our

_proof

follows Laurent’s

_original

idea

_[11].

In Section

_2.2,

we consider

the real-valued determinant of a certain

_matrix;

this is akin to Baker’s

auxil-iary

function but

_(as

in the case with Schneider’s

method)

avoids

"multiplic-ity"

estimates caused

_by

derivatives. In Section

_2.3,

we use the Maximum

Modulus

_Principle

to obtain an _upper bound on this determinant

(assuming

that our non-zero linear

_{form-actually}

a modification thereof-is "close"

to

_0).

In this we

_improve

the estimates of

[33,

_Chapter

9]

in the case of

three

_logarithms.

In Section 2.4 we

_give

a lower bound on the absolute

value of the determinant

using

an

_inequality

of Liouville

(assuming

that

the determinant is

_non-zero).

In Section

_2.5,

we combine these results to

obtain a contradiction

( to

a

modified

linear

f orm being

close to

₀₎

under

appropriate assumptions

on certain _parameters.

Consequently

we deduce

that our modified linear form is not too close to 0.

In Section

_3,

we take fuller

_advantage

of the

_theory

of

_algebraic

varieties

in

_3-space.

We

_considerably

refine the

_{general theory}

of "zero" estimates in this

_{setting by}

a careful

_analysis

of the intersections of translates of

hypersurfaces

under the group

₀₂₊

x

Q:.

Our contribution here is to use sets

of

_different

cardinalities to lower the dimension of certain

_algebraic

sets and

to eliminate one

key

case

by

a combinatorial

_argument.

As a _consequence

of this

_analysis,

we are able to deduce that our determinant is non-zero

under

considerably

weaker

_hypotheses

than had been used

_previously.

In Section

_4,

we determine constants that fit all our

_required

specifica-tions,

and in Section 5 we show how these result in

bounding

our

origi-nal linear form

_sufficiently

_{away from 0.}

_Finally,

in Section 6 we

_provide

computer-generated

tables to

_complete

the

_proof

of Theorem B

_(and

so

Theorem

_A).

These tables will be useful for anyone

wishing

to solve

Dio-phantine

equations

that lead to linear forms in three

_logarithms.

Acknowledgements:

It is a

_great

_pleasure

to thank Alan Baker for

_greatly

encouraging

us to _pursue this line of research

_specifically

for three

loga-rithms and Michel Waldschmidt for

_having

_{previously provided}

us with a

preprint

of

_[33]

which made it

_possible.

We are also

_grateful

to Michel

Waldschmidt for

_finding

a "howler" in an earlier version of this _paper, and

We have been fortunate to receive the

_support

of both

_throughout.

We also

owe a

_great

thank you to Paul Voutier for the many hours he

spent

check-ing

our results. We are

_especially

_grateful

to him for

_pointing

out a more

felicitous

_proof

of Lemma 2.2 for the case A

_0;

our

_{original proof}

resulted

in an increase in the constant in the table

_by

a factor of

approximately

2

when A was

_negative.

Sl.

_Background

Definitions and Lemmata.

Throughout

the _paper we will use

IAI

to denote the

_cardinality

of a set

A. We will also use

[a]

to denote the

_greatest

_integer

less than or

_equal

to

a real number a. So

[a]

a

[a]

+ 1. 1.1. A Combinatorial Fact.

We will need the

_following

combinatorial result from

_[14]

_(where

their K

corresponds

to our

K2,

and their S to our

ST):

LEMMA 1.1.

_[14,

Lemme

_4].

Let

_{K, L, R, S, &#x26;T}

be

_positive

rational inte-gers and N =

K2L.

Let In

=

~(n - 1)/K~~(1

n

N)

and

(rl, ..., rN)

be

a _sequence of

_integers

belonging

to the set

_10,

.. ,

R -1}.

Suppose

further

that N

_RST,

and for each r E

_{{0, ... ,}

= ST. Then

1.2.

_{Algebraic Geometry.}

Throughout,

if K is a

field,

we use

K[X,

_Y,

Z]

for the

_ring

of

_polynomials

(over K)

in the

_commuting

variables

_{X, Y, Z.}

If P C

_K[X,

_Y,

_Z],

we write

for the

_variety

_generated

_by

_{P, i.e.,}

Any

such set is also called an

_algebraic

set.

We will need some

_background

from

_algebraic

_{groups. We will} confine

our attention to

_subgroups

of G =

Q+

x

_Q+

x

_Q*,

where the

_operation

(denoted

by

+)

is addition on the first two coordinates and

_{multiplication}

A

_subgroup

of G that is an

algebraic

set

_(the

set of common zeros of a

_family

of

_polynomials

_{{P(X,Y, Z) : P E P} )}

is called an

_{algebraic subgroup. Any}

algebraic

subgroup

of G has the form U x

(*,

Lf x

{1}

or Lf x for

some vector

_subspace

of

_Q2

and

_{positive integer}

m, where

Tm

is the

multiplicative

group of

m’h

roots of

unity

[9].

For

simplicity,

we write

To

for

_{~"‘ .}

An

_algebraic

set ~S is said to be irreducible if it cannot be written

as the union of two

_algebraic

sets

_properly

contained in S.

_Every

_algebraic

set can be written

uniquely

as a finite union of irreducible

algebraic

sets

_[9,

Corollary

I.1.6~,

and these constituent irreducible

_algebraic

subsets of S are

called its irreducible

_components.

It is

_possible

to define the dimension

_of

an

irreducible

_algebraic

set so that

_points

have dimension

_0,

_Q3

has dimension 3 and if

_{V, 5}

V2

with

_{Vi, V2}

irreducible

_algebraic

_sets,

then

_{dim VI}

dim

_V2

_(see,

_e.g.,

_[10,

_Chapter

_2]).

These definitions coincide with the usual intuitive ideas: irreducible curves have dimension

_1,

irreducible surfaces

dimension 2. The dimension

_of

an

_algebraic

set is

_just

the maximum of the dimensions of its irreducible

_components.

In this

_setting,

we define the

_bidegree

of a

polynomial

P(X,

Y,

Z)

E

Q[X, Y, ]

to be at most

_{(Do, DI)}

if

_{P(X, Y, Z)}

= with

= 0 whenever i

+ j

_&#x3E;

Do

or k

&#x3E; Dl.

Let

de-note the set of

_polynomials

in

_{Q[X, Y,}

_Z]

of

_bidegree

at most

_{(Do, Dl).}

For any subset E

of Q

x

Q,

let be the set of all maps from E to

Q.

Consider the

_Q-linear

map

which maps each P E

Q[X,y, Z]

to the restriction to E of the

_polynomial

map from

Q3 to Q

induced

by

P. For each

pair

of

non-negative integers

(Do, Dl), let

If

E

is the smallest

_algebraic

set

_containing

_E,

then as _reSE and _resË have

the same

_kernels,

we obtain

If

Do

and

_Dl

are not too

_small,

then

_H(E;

_Do,

_{Di )}

coincides with the value

This

_polynomial

is called the Hilbert-Samuel

_{bihomogeneous polynomials of}

E. We denote

_by

_l-t(E;

_Do,

_D1)

the

_product

of n! and the

_homogeneous

part

of the Hilbert-Samuel

_{bihomogeneous polynomial}

of E of total

_degree

n evaluated at

(Do, D1).

So

We will

_require

the

_following

facts: LEMMA 1.2. With the above notation:

(i)

(ii)

Let U be an

_algebraic

subset

Q2

of dimension and m be a

non-negative integer.

If the

_subgroup

_Tm

_{of (Jt}

has dimension then

(iii)

If V is a

_non-empty

_algebraic

subset of G

_{and g}

E

_G,

then V _{+ g} is an

algebraic

set of the same dimension as V and

(iv)

If H is an

algebraic subgroup

of G and E is a finite

_non-empty

union

of translates of H in

_G,

then E is an

algebraic

subset of G and

(v)

If v and X are

_algebraic

subsets of G and E =

Ig

_E

G : g

+ 1~ C

X},

then E is an

algebraic

set.

_Moreover,

if X is defined

_{by polynomials}

of

bidgree

at most

_{(Do, D1 ),}

then so is E.

(vi)

If E is a

_non-empty

algebraic

subset of G defined

by polynomials

of

bidegree

at most then

Part

_(i)

is obvious

_(see

_[33,

_Chapter

_8]).

Parts

_(ii)-(vi),

are

1.3. The

_Multiplicity

of a Zero.

Let

Ko

and I be

_positive

_integers

and

_{4&#x3E;1,... , ~7}

be

_analytic

functions in C. Let

_{(l~z, ~ni) ~1 i}

_{I )}

be

_pairs

of

_non-negative

rational

_integers

whose

sum is at most

_Ko.

Let

_{il, t2 E}

C and for each i E

_{~1, ... ,}

_)7},

let

Further let

_{(j) :}

_{1 j}

_{I }}

be a set of elements of

(C 2 .

Let N denote the set of

_non-negative

rational

_integers

and for each

(k, m)

E

N 2,

let

_{11 (k, m) 11}

= k + m. We define

8(Ko,I)

_to be the minimum

value of as

«k1,ml)... (k1, mz))

_ranges over all

I-tuples

of elements of

N2

which are

pairwise

distinct with _{ml , ... ,}

Ko .

LEMMA 1.3.

_[33,

Lemma

_9.1].

The function of one

_complex

variable x

given by:

has a zero at x = 0 of

multiplicity

at least

8(Ko,I).

We now

slightly modify

the

proof

of

[33,

Lemma

9.2]

and

_get

an

improved

lower bound on

LEMMA 1.4. Let and I be

_positive

rational

_integers

with

_{Ko &#x3E;}

998, L &#x3E; 18,

and

_(Ko

+

_{2)2L/4 &#x3E;}

1 &#x3E;

_{.lSKa L.}

Then

Proof :

The smallest value for the sum + ... + is

reached when we choose

_{successively,}

for each

_{integer n}

=

_{0,1, ...}

all the

points

in the domain

Hence the number of

_points

that we

_get

_{by varying n}

between 0

and,

say, A - 1

_{(with A &#x3E; Ko )}

is

Moreover,

if A is maximal such that this number is at most

_I,

then

_ _, 1 a _,

Now A is maximal such that Note

_that,

as

desired,

Finally,

we need to show that

Since the function

_{J(x) =}

_{( x~)x - 3L ( Ko+1 )2}

is

_decreasing

as x increases

in the indicated range

(x &#x3E;

(.15)Ko2L),

we need

only

check the truth of

the

_inequality

at the maximum value of I. Our

_inequality

holds when I =

(Ko

₊

2)2L/4

provided

that

Since

_{Ko &#x3E;}

_998,

the

_righthand

side is at most

_(.239)L

+

_k.

Hence the

inequality

holds if

_{(.011)L &#x3E;}

_k.

Our

_assumption

that L &#x3E; 18

_guarantees

this. D

1.4. A Liouville

_Inequality.

We will need the

_following

_special

case of an

inequality

that dates back

to J. Liouville.

_(For

a fuller

_account,

see

[33,

_Chapter

III]).

LEMMA 1.5.

_[33,

Lemma

_3.14].

Let be non-zero rational

num-bers;

say a~ = with ajl,

coprime

rational

_integers

(j

=

1, 2, 3).

=

1, 2, 3).

Let

f (X,Y, Z)

_e

be such that 0.

_Ki,

_K2

and

K3,

then

1.5. A Product of Factorials.

The _purpose of this section is to derive a lower bound for

(k!).

The

proof

we

_give

was

kindly suggested

to us

by

Paul Voutier.

LEMMA 1.6. For any

integer K &#x3E;

6,

we have

Hence

By

the Euler-Maclaurin Summation

_Formula,

Therefore

Hence,

in order to _{prove the}

_lemma,

it suffices to show

_(for

K &#x3E;

₆₎

that

This will

_certainly

hold if

S2.

_Analytic

Results. 2.1.

_{P reliminary}

Definitions.

Let be

_{multiplicatively independent}

rational numbers

_greater

than

_1;

_{let ai = with ajl,aj2}

coprime positive

rational

_integers

(j

=

1, 2, 3 ) .

and _assume

(without

loss of

gen-erality)

that a3. Let

bl,

b2,

b3

be

positive

rational

integers

with

gcd(bl, b2, b3 )

= 1. Let A =

b2 log a2 -

bl log al -

b3

log a3 ~

0 with

b2 log

a2 =

maxf bj

log

aj j

= 1, 2, 3}.

Let K and L be

_positive

_integers,

and N =

K2 L.

Let i be _an index such that mi,

li)

runs

through

all

triples

of

_integers

with 0 K -

_1,

So each

_{number 0, ... ,}

K - 1 occurs

KL times as a

_~Z,

and

_similarly

as an _mi; and each number

_{0, ... ,}

L - 1

occurs

K2

times as an

_~.

Let

_R,

S and T be

_positive

rational

_integers

with RST &#x3E; N =

2.2. The

_{Interpolation}

Determinant.

The rest of Section 2 is an

_adaptation

of Laurent’s method of

interpola-tions

_(see

_[13]

or

[14]).

With the above

definitions,

let

where

_{Sj, tj}

are

_{non-negative integers}

less than

_R,

_{S, T,}

_{respectively,}

such that runs

_through

N distinct

_triples.

Suppose

that these N distinct

_triples

can be chosen so

_{that A #}

0. Let

b1, b2, b3

E

_Z+,

and let

_{31

=

bi /b2,

_~3

=

b3/b2.

Recall

Let

where

and

Since

_{s;, b2, L}

and

_{I A}

are all

positive,

I

A/2

S,

we have

where x = Now decreases in

(0, oo)

and tends to 1 _as

x --&#x3E;

0+

_{by L’H6pital’s}

rule. Hence

10ij (

1.

We wish to bound A away from 0. As a first

_step

we bound A’ _away from

0. Under

_appropriate

_assumptions,

we now show that A’ &#x3E;

1/ pKL,

for an

appropriate

p &#x3E;

e.

(We

will use the

_Proposition

2.1 to show a similar result

for

_IAI

in Section

_5.)

PROPOSITION 2.1.

_Suppose

that K &#x3E; 500 and L &#x3E;_ 20. With the above

0,

then A’ &#x3E;

1/ pKL

_provided

We will establish

_Proposition

2.1

_by

reductio ad absurdum. We will first obtain an _upper bound on

~

[ (assuming

that A’

1);

the

_key

tool will

be an

_application

of the Maximum Modulus

_Principle

and the results in

Subsection 1.3. We will next derive a lower bound on

~0~

(assuming

that

A 0

0)

using

Liouville’s

_inequality

_{(Subsection 1.4);}

_however,

_assumption

(2A)

forces the lower bound to _{exceed the upper bound. This contradiction}

establishes the

_Proposition.

2.3. The

_Upper

Bound for

_JAI.

In this

_subsection,

we _prove:

and

_similarly

for

_{( tj b2+Sj mi b1).}

_Hence,

_using

the

_{multilinearity}

of

determi-nants,

we obtain that

By

(2.2), (2.3),

and the definition of

_{A; ,}

it follows that

Since

EN- 1 Ai

=

_0,

_we deduce that

where

N

=

{0,1, ... , N -1 }

and

AI

is the determinant of the matrix

whose

_{(i, j )}

_entry

is

where

Let

Then,

letting

w¡(x)

=

_gives

By

the Maximum Modulus

_Principle,

if were

_analytic,

then

xii

,-- --I

Define

G1

=

LRN/8,

G2

=

LS’N/8

and

G3

=

Now,

by

Lemma 1.1

and the fact that

_Ài

=

0,

we obtain

Similarly

_{~r(~) ~}

~3+~302.

Since 0152l and a3 exceed

1,

we deduce

Using

(2.7),

the

_equality

the definition of

_b,

and that

2.4. A Lower Bound for

_~.

Under the

_assumption

_{that A #}

_0,

we use Lemma 1.5 to establish

LEMMA 2.3. With the above

_assumptions

and

_notation,

Proof:

Consider the

_polynomial

where ,,

and a _ranges over tne elements ot tne _{symmetric group} on lv _letters; as

usual,

denotes the

_signature

of u.

Clearly A

=

P ( al , a2 , a3 ) .

_By

Lemma

1.1,

Hence

Now

and since ’ I

Consequently,

(2L)

follows at once from

(2.8).

0

2.5.

_Synthesis.

By

(2U)

and

_(2L),

we deduce that

This

_simplifies

to

Since

_{log aj}

&#x3E;

_{0 ( j}

=

1, 2, 3 ) , substituting aj

for aj and

_using

the definition of

_{G~ ( j}

=

1, 2, 3),

we

_get

a contradiction to the

_assumption

~A~~

₁

if

_inequality

_(2A)

holds. This establishes

_Proposition

2.1. 0 Our next

_goal

is to

_put

conditions on

_R,

_S,

_T,

K and L to ensure N =

K2 L

distinct

_triples

can be chosen so that the

resulting A

is non-zero.

Equivalently,

so that the N x RST matrix

_resulting

from all

_triples

has full

row rank.

_By

row

_reduction,

as in

(2.2)

with _qo and

~o

both

replaced by

0,

this occurs if and

only

if the matrix with

(i, j)

_entry

(r~

+ +

has full row rank. That

_is,

if and

_only

if : the

_only

polynomial

EiÀixkiymizli

having

all

_tj+Sj33,

as

roots

_{(0 r~}

_R;

0 _s;

_S;

_{0 t~}

_{T )}

is the zero

polynomial.

This concerns

polynomials

with X and Y

degree

at most K -1 and Z

degree

at

most L - 1.

S3. Row Rank.

In this section we _prove a result about zeros of

polynomials

that

guar-antees that the full row rank is attained

_(except

under

_propitious

cir-cumstances when better results are

possible).

Here is where we

_heavily

use

_algebraic

_{groups. Our} zero estimate is similar to

_{Philippon’s}

Zero Estimate

_([33,

_Chapter

_8])

but

_requires

a close

_analysis

of the intersec-tions of translates of

_{hypersurfaces}

under the group

_q+

x

Q:.

It allows

us to both eliminate the need for

_E3

and take

_El

and

_E2

to be sets of very different sizes. We will establish

(under

mild

conditions)

that either satisfies a linear

_dependence

relation with small

_integer

coef-ficients,

or the

only polynomial

of

_prescribed

bounded

_{degree having}

all

(r~

+ +

_s~~3,

_{(j E}

_{10, -}

)~ 2013

₁₁₎

as zeros is the zero

polynomial,

where

_/3~

=

(j

= 1,

2, 3).

Specifically:

We define

b2,

b3}

to be

_(R,

_S,

_T)-linearly

_dependent

over Z if there

exist E Z not all 0 such that

_dlbl

+

d2b2

+

_d3b3

= 0 with and

_{Id11, Id31 ~}

_S;

and if

d2

=

0,

then

Id11 ~

T and

Id31 ~

R.

THEOREM 3.

_Suppose

that

_{K, L, R,}

_{S, T, R1, Si,}

_T1,

_{R2, s2,}

T2

are

_positive

integers

greater

than or

equal

to

_3,

and that K &#x3E;

_{2L, Rl}

+

R2

R,

S,

+

S2

S and

Tl

+

T2

T. Further assume that

R1

without loss

of

_generality.

IF

and

THEN either

_b2,

_b3}

is

_(R,

_S,

_T)-Iinearly

_dependent

or the

_only

polyno-mial

_P(X,

_Y,

_Z)

E

_Y,

_Z]

_{(with degx}

_P,

_degy

P K -1 and

_{deg z}

P

L -

₁₎

_{having {(r}

+

_S/31,t

+ 0

_r

_{R - 1,}

0 s

the

K2 L

X RST matrix with

(i,j)

_entry

tjb2+-Sjb3)a

ki

)(

mi l 1

2’

has row rank

_equal

to

K2 L.

_{Consequently,}

for an

_appropriate

subset

of K2 L

columns,

the

_resulting

matrix has determinant A that is non-zero.

Under condition

_(2A)

of

_{Proposition 2.1,}

this will

_yield

our lower bound

for A’.

To prove Theorem

3,

we _argue

by

reductio ad absurdum. Let

_{~j =}

Without loss of

_{generality, P(X,}

_Y,

_Z)

can be written as a

product

of

dis-tinct irreducible

_polynomials

PI, ... , Pn

(none

of which is a scalar

_multiple

of

_Z).

Let

.~o

=

V(P)

=

Uf V(Pi) :

:1 z

nl,

the set of zeros of P. Let

(Zo - (1) =

(Y(Pz) - (1).

Letting

(A1, ... ,

An)

range

over all

_n-tuples

of pairwise disjoint

subsets of

El

whose union is we see

(by

DeMorgan’s

Laws)

that

_{Zl =}

_{o~) .}

Note

_that,

_by

_hypothesis,

E2

C If is not

_{(R,8,T)-linearly}

dependent,

then the

_projection

of

_Ej

onto the first two coordinates is a

one-to-one _map

_{(j =1, 2);}

_i.e.,

if tj =

0 r 0

~ ~, 0 t T,}, then ~ = (R, +1)(~ + 1)(T, + 1)

(j

=

1,2).

Throughout

we let

_Do

=

2(K -1)

and

D, =

L -1,

and assume

(without

loss of

_generality)

that

Ri T1.

We wish to establish that the

_non-empty

_algebraic

set

,~1

has dimension 0. This will follow from a series of lemmas in the next section:

3.1. The dimension of

,~l

is 0.

If

dim Zi

&#x3E;

_0,

then for some

partition

f A,,... ,

of we must have

d.imCi

&#x3E; 0 for i E

_{{I, ... ,}

_n~,

where

Ci

=

n(7’EAi (V(Pi) -

(1).

PROPOSITION 3.1.1. Under the

_hypotheses

of

_{Theorem 3,}

the number of elements

_lying

on a common tine is bounded

_by

unless

_{{&#x26;i,&#x26;2?&#x26;3}}

is

_dependent.

Proof: Suppose

the number of elements of

_t,

_lying

on a common line i of

slope

m is

_greater

than M and that is not

_{(Ri , Si , Ti )-linearly}

lying

on i such that

Simple slope

calculations now

_imply

Case A: m oo. We shall further assume that _{7i,... a4} are chosen

such that

_{182 -}

_I

is minimal with

_respect

to ri = r2 and that

[r4 - r3 ~ [

is minimal with

_respect

to S3 = s4. Since m =

finite,

it

2 11 1

follows that

_It2

-

I

is also minimal with

_respect

to rl - r2 .

Similarly,

lt4

-

t3

is minimal with

_respect

to s3 - s4. We have

Clearing

denominators and

_remembering

that

_/31

_{- b}

and

_/?3

=

~,

we

obtain the

_equation

b2 b2

By

the

_minimality

of

_{s2 - sx in}

the choice of ~1 and 02? we know that for

each r with 0 r

Ri ,

there are at most

s 1 sl

different s’s 2 1

such that there exists t E

_{0,...

with

_(r

+

8/31, t

+

_{s,Q3 )}

E .

_However,

since m _oo, there is at most one such t for _every

_pair,

_{(r, s).}

_Hence,

all

_together

there are at most

2R+.)CSi+1)

_points

of

_{Ei on}

.

_By

our

182-811

Exchanging

the roles of r and s in the above

_argument,

we also obtain that

(since

Rl

Ti)

and

Since

_{m #}

_oo, we have _{r4 -}

_{r3 ~}

0. If _t2 =

_tl,

then we have

We deduce that if

_{m ~}

oo

(and

hence _{s2 -}

0),

then is

(R1,

Sl,

Tl )-linearly

dependent.

This contradiction arose from our

assump-tion that the number of

_points

of

_Ei

_lying

on l is

_greater

than M. Case B: If the

_slope

were

infinite,

we would then have

Hence

Therefore

b2,

b3}

is

_(Rl,

_Sl,

_Tl)-linearly

_dependent,

thus

_proving

Propo-sition 3.1.1. D

If

Pi

has

_bidegree

_{(1, o),}

for each E

_Ai,

Q is a line. Since

1 and

Ci

=

no-EAi (V (Pi) -

~),

all _{u E}

Ai

lie _on the _same translate

of the line

_Hence,

_{by Proposition}

_3.1.1,

= M. Let

K = where A =

{z

_E

~1, ... , n} :

_Pi

has

bidegree

(1,0)}.

Then

-

.,..--Since P has

_bidgree

at most

_{(Do, Di),}

we

_{Do .}

_Further,

since

I E, I

&#x3E; ~M

_{(by (3.4)),}

the

_complement

of A is

_non-empty.

We note an

essential difference between the elements of A and its

_complement.

Write for translation

_{by cr}

E

_G;

i.e.

_(X

+ +

_{2? 3)}

where

LEMMA 3.1.2. Let

_{P2 (X, Y, Z)}

be an irreducible

_polynomial

other than

_Z,

and E

~1

7’. =

then Pi

has

bidegree

(1, 0).

Proof :

If

_{V (Pi}

o

tu) = V (Pi

o

to’,),

then

Pi

o

_to’

=

cPi

o

_t,,

for some c E

~* .

We first show that

_{degz Pi}

= 0.

Write

(7 =

(r

+ +

_~/?3,~~~)

_(r’

+ +

s’ Q3 , al’ ag’ a§ ) .

Examining

the term with l maximal so that

~

0 for

_{some j, k}

and

j

+ 1 is maximal

_subject

to l

_being

maximal

_(if

there is more than one such

( j, k ),

choose _any of

_them),

we must have

and hence c =

(a’-"

)1.

Next _we examine the coefficients of the

highest

order term

_{of Pi}

with _{Z power}

_equal

to 0. That

_is,

we look at the

coefficient maximal with

_respect

to 0. If no such term

exists,

then we have that Z divides

Pi

_contrary

to our

hypothesis.

If more

than one such term

_exists,

choose _any of them. In this case, we obtain that c = 1. Hence

I I ..1 .

Since is

_{multiplicatively independent,.e}

= 0. Therefore

degz

Pj

= 0.

We now shew

that Pi

is indeed linear in X and Y.

Since Pi

if But

degz Pi

=

0,

_so

(xo, Yo)

+ Z (ui - u[ , u2 - ~2 )

(where u =

(~1,0-2,~3))

is contained in the

_variety

_Wi

of

_Q2

defined

_{by Pi}

_(viewed

as an element

of

_{Q[X, Y]) .}

Since

_Wi

_passes

_{through infinitely}

_many

_points

of the line

(xo, Yo)

+

_{32 2013}

we obtain

_by

Bézout’s Theorem

[9,

Corol-lary

1.7.8]

that

Wi

is this line.

_{Hence Pi}

has

_{bidegree (1, o)}

as claimed. 0

Our next task is to bound

_{IAi if i ~}

_{A. So suppose that}

_{i ~}

A is fixed.

Assume that 2. We now show that

Ci =

u)

is

one-dimensional. Let

_11i

=

{T1, ... ,

and

_{(for j}

=

_{1, ... ,}

ni) 1-(,j

denote the

hypersurface

V (Pi

o

_{tT~ ) .}

_By

_{Lemma 3.1.2} we have that the

_{hypersurfaces}

?C 1

and

_1-£j2

are not

_equal

_{whenever jl}

_{j2 , j 1, j2 E 1, ... ,}

There-fore,

n

_{x2 )}

1. Since

dim Ci

&#x3E;

_0,

we must in fact have that

has dimension

_1).

Thus 1. It follows from the

_{generalization}

of B6zout’s theorem

_([9,

Theorem

_1.7.7])

that

_{?ti1 n ?ti2}

has at most

_N2

irre-ducible

_components,

where

Ni

=

deg

Note that

Do + Dl - /1,.

We will now establish three lemmas which will allow us to bound

_IAi/.

LEMMA 3.1.3. Assume all of the above

_hypotheses

and notation.

_{If Ci}

is

a

_component

of1-li1

n n n

?-~j4,

and Tj2 - Tj, - Tj4 -’

(with

Tjl, ... , Ti4 E

_Ai),

then _-,rj

(Ci)

is a

component of1ti¡

n

1ti2.

Proof:

Let v E

Ct.

_{Since Tj 1, ... , Tj4} E we know

From these

_equations,

we deduce

yielding

that _-,,,

_(v)

E

_{?~~1. Similarly}

and hence also.

_Therefore,

C ₁

_{n 1tj2’}

implying

that

_{t, (Ci)}

is a

_component

of n

_1-ij2.

This proves Lemma 3.1.3. D

LEMMA 3.1.4. With the above

_assumptions

and

_{definitions, if a, a’}

E

_Ai

and

_{tcr( Ci)}

= _tr,

(Ci),

= (1’.

Proof :

Let F =

~g

_E G :

Ci

₊

g = which is _an

algebraic subgroup

of G

by

Lemma

_1.2(v).

Hence F = Ll

xTm

for _{some vector}

subspace U

of Qf

and

m &#x3E; 0. If

_(Ci)

=

(Ci),

then _{a -} u’ _E F.

If a 54

~’,

then _as

(ai ,

cx2,

a3 ~

is

_{multiplicatively}

_independent,

it follows that m = 0.

Moreover,

since the

projection

of

_~iZ

onto the first two coordinates is

_one-to-one,

the

_projection

of ~’ is a non-zero element of

_U;

so dimU &#x3E; 1. Hence

_2,

a

LEMMA 3.1.5. With the above notation and

_hypotheses,

the set

_~12

cannot

contain

_N2

-f- 1 distinct

_pairs

such that is a non-zero

constant

_independent

_{of j .}

Proof :

Suppose

otherwise.

_By

Lemma

_3.1.3,

for each such

_pair,

_-Tll

_(Ci)

is a

_component

of

_1tjl

₁ n

_1-(,j2.

Since we know

that 1-(,jl n 1íj2

has at most

components,

the

_pigeonhole

_{principle implies}

either for

_{some j ;}

1 we

have

_tTjl-Tl1

_{(~Z) -}

or there

_{exist j}

and j’

such that

In the former case, we have that =

(Ci),

and in the latter _case,

we have

_{tril (Ci)}

=

tT.,

(Cz).

Since

if j j’, we

_get

a contradiction

31 ₃₁ 1

to Lemma 3.1.4. Hence Lemma 3.1.5 is true. 0

We now have the tools to bound To do _so, we will count the total number of

_possible

differences for the

_pair

Then we will count

the number of distinct

_pairs

of elements of the set

_Ai.

_{By comparing}

these and

_using

the

_pigeonhole

_principle,

we will be able to find a bound.

Recall Since 0 ~ r

_Ri,

0 s

_{Sl ,}

and 0 t

_{Ti ,}

the number of

_possible

non-zero values for _{7j2 where} E

_~1z

is

at most

On the other

_hand,

the number of distinct ordered

_pairs

of distinct elements of

Ai

is

_{1 ) .}

Thus if

we would obtain a contradiction to Lemma 3.1.5

_by

the

_pigeonhole

princi-ple.

Hence,

Applying

the

_{quadratic formula,}

we have:

This is the bound we want for Since

Do

+ and

n - K

)Do

+

Dl

(since

there are at most

_)Do

factors with

_Z-degree

0

not

_{having bidegree}

(1,0)),

we have

Hence,

by

(3.1.1)

and

_(3.1.2),

and 0 K

_Do.

This

_{right-handside}

is

_clearly

maximal when = 0. Since

and &#x3E;

8(2K+L-2)2,

inequality

(3.1.3)

is

_impossible.

Hence

dimZ1

= 0. 0 3.2. Proof of Theorem 3.

We first show that

_(under

the

_hypotheses

of Theorem

₃₎

there exist a

vector

_subspace

Lf of

_Q2

of dimension 0 and a

_subgroup

of

Q*

of

dimension

₆₁

with

₆₀

+

₆₁

2 such that

Proof:

Write

Do

for

_{2(K -}

₁₎

and

Dl

for L - 1. Let

_{H(E, Do, D1)}

be the

_homogeneous

_part

of

_p!

times the Hilbert-Samuel

_polynomial

of total

degree

p

(= dim E).

By

Lemma 1.2

(i)

&#x26;

(ii), 7-L(G, Do, Dl)

=

3DoD1

and

+

_H(U

x Therefore it suffices to

show that

for some

_{H ~}

G an

_{algebraic subgroup}

of G

(then

H =1,t x

T,).

For each

~

E 0

x Qx ~*,

let

(X

where

7rl, Jr3 denote the natural

projections

onto the

respective

coordinates.

We will for when Z

_{C 0 x Q}

x

~*.

Let

_.~o

=

V (P)

denote the set of _zeros of

P,

and

Zi

=

(1),

the set of common zeros in G of the

polynomials

P o

_(U1

_E

_{£i) ,}

_each

have

E2

C Let

Z2

=

~2)

C

21.

Then 0 E

Z2,

and

Z2

is an

algebraic

subset. Since

dim Zl

=

0,

dim

,2

=

dim 21.

Let

Vo

be a common irreducible

_component

of

_.~1

and

_Z2

of this dimension. So

vo g

~2 =

Thus for all T E

E2,

we have T +

Vo

C Let

Let H = (g E G : g + Vo = Vo)

and

_Xo

=

{g

₊

Vo :

g E E}.

Then

Vo

and each of the elements of

Xo

are

algebraic

subvarieties of the same dimension as

Z,

(by

Lemma

1.2(iii)).

They

are also contained in

_,~1

and so

_Xo

is finite.

Clearly

H is a

_subgroup

of G and E is closed under

_th

_{(translations}

by

h)

for all h E H. Hence there is a

_bijection

E/H ~

Xo.

Thus E is a finite union of translates of

H. Now H is an

algebraic

subset and so an

_{algebraic subgroup}

of G. Since

0 0

Vo ~

G,

we have G.

_By

Lemma

1.2(v)

E is an

_algebraic

subset of

G

_which,

like is defined

_by

_polynomials

of

_bidegree

_{(Do, Dl).}

Since E is a finite union of translates of

_H,

Lemma

1.2(iv)

_gives

Since E is defined

_{by polynomials}

of

_{bidegree (Do, Dl ),}

we have

_(by

Lemma

_1.2(vi))

_1i(G,

_Do,

_Dl).

_Finally,

since

E2

g

E we

have

This establishes the claim. 0

Now H = Lf x

Q* (respectively

Lf x =

{g

_E G :

Vo

_{+ g}

=

Vol

as m = 0

(or

m #

0)

for some vector

_subspace

Lf of

_Qf

of dimension

_bo,

where

Vo

is a common irreducible

_component

of

_Z,

and

_Z2

of dimension 0.

{0}

and m =

_0,

and

{(xo, yo, zo)}

=

_Vo,

then

(xo, yo, z)

_E

1Jo

for all z E

_{~* .}

So

_1,

a contradiction.

_

IfU #

{0},

let E

_{U) ( 0 )}

and Then

Vo.

Thus dim

_Vo

&#x3E;

_0,

a contradiction. Hence H =

{0}

with m 0 0.

Now if

_{(xo, yo, zo) E}

_Vo,

then

_Vo.

_{Consequently,}

m =

1,

and H is the trivial group. Thus

(3.2.1)

becomes

(R2

+

1)(SZ

₊

1)

S4. Numerical Constants.

In this

_section,

we follow standard

_procedures

to obtain numerical

con-stants that

_satisfy

the conditions of both Theorem 3 and

_Proposition

2.1. Let _{Cl, C2, C3} &#x26; c4 be real numbers

greater

than

_1;

and _suppose that

and assume that

( B °

&#x3E; ) B &#x3E;

max( Bo , log

b’ .

We will assume that

Bo

&#x3E; 10.

Let

.,....,. r , ,

Note that is

_(R,

_S,

_T)-linearly

_dependent

over Z if and

_only

if it

is

_{(C4, B)-lineaxly}

_dependent

over Z

(with

_respect

to

_(a,,

_a2,

_a3)).

Let

PROPOSITION 4.1. With the above

_definitions,

condition

_(2A)

holds as do

the

_hypotheses

of Theorem 3

_provided

that

and

where

Bo &#x3E;

10 and _{e p}

_9,

and the two terms

_involving

B°

are omitted

if B°

is not known.

Proof :

We first show that b b’

_(provided

that

_{e3~2c4 )}

and hence

By

definition,

Hence it suffices to _{prove that}

By

Lemma

_1.6,

the

_right

hand side exceeds so the desired

inequality

certainly

holds if

_c4

That the

_hypotheses

of Theorem 3 hold

_(under

suitable

_assumptions

on

the constants cl ... c4 for these values of and

L-see

below)

follows

by

grubby

school arithmetic.

Specifically,

.K &#x3E; 2L since

2c2.

Moreover,

since c3 &#x3E; 1 and 1

_{( j}

=

1, 2, 3),

we have that

conditions

_(3.1)

and

_(3.2)

both hold

_provided

that

Since B &#x3E;

_{Bo &#x3E; 10,}

this is

_obviously

the case.

Inequality

(3.4)

certainly

holds

_provided

that

Now

_C2/ci

_.0002,

so we

_{merely require}

that

_c3

&#x3E;

2(2.0002)~c~.

So we take

Note also that c3 &#x3E;

_3,

so 3.

Next we want

R2 &#x3E;

3. For this we need

_only

insist that R &#x3E;

R1

+ 4. Since 1

_{( j - 1, 2, 3)}

and B &#x3E;

_Bo,

it suffices that

Similarly

S2 &#x3E;

3 and

_{T2 &#x3E; 3,}

_assuming

_(4.3).