C OMPOSITIO M ATHEMATICA
W. D ALE B ROWNAWELL
On the Gelfond-Feldman measure of algebraic independence
Compositio Mathematica, tome 38, n
o3 (1979), p. 355-368
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355
ON THE GELFOND-FELDMAN MEASURE OF ALGEBRAIC INDEPENDENCE
W. Dale Brownawell*
@ 1979 Sijthoff & Noordhoff International Publishers-Alphen aan den Rijn Printed in the Netherlands
In
1950,
A.O. Gelfond and N.I. Feldman established their well- known measure ofalgebraic independence
ofa(j
anda(j2, where
a # 0 isalgebraic, log
a -:;é 0and 8
is a cubicirrationality [8]:
The numbers
a 0
anda s2
had been provenalgebraically independent by
Gelfond[7].
The totaldegree
of apolynomial
P will be ab-breviated as
dp
ordeg P
and thelogarithm
of itsheight
ashP
orlog
ht P.In Theorem 1 of this paper we
improve
the lowerbound - exp(t4+E) to - exp(Cd’ P (dp + hp».
We obtain this result from theproof
ofTheorem 2. But in contrast with the theorems of Gelfond and Feld-
man and even the recent
generalization
of Gelfond’s result ona/3, a by
M. Waldschmidt and the author[4],
where a is taken to bewell-approximated by algebraic numbers,
the arithmetic nature of aby
itself is notspecified
in Theorem 2.However,
as in[4], [2],
bothlower and upper bounds on
expressions
in thèse numbers areused,
even when no transcendence measure for any of them is known. This basic idea comes from G.V.
Choodnovsky’s proof
of thealgebraic independence
of three numbers[5]
based on Gelfond’s method. The result of[2] applied
to more instances than did the earlier lowerbound of Gelfond and
Feldman,
but was weaker than it in the casethey
considered.* Research supported in part by the National Science Foundation.
0010-437X/79/03/0355-14 $00.20/0
Since then G.V.
Choodnovsky
has established thealgebraic
in-dependence
of n + 1algebraically independent
numbersby
a remark-able iterative
application
of a variant of Gelfond’s method. There itwas necessary for him to
develop
a useful extension of the concept ofa resultant to
polynomials having
a common factor(see
Lemma 5 in Section 3below).
Weemploy
these so-called semi-resultants to prove a theorem on a sort of simultaneousapproximation problem
which willgeneralize
andsharpen
somewhat the result of Gelfond and Feldman.(Actually
thesharpening
is obtainable from theoriginal
structure of
proof
aswell.) Choodnovsky
himself has announced astrengthening
of the Gelfond-Feldman result to a lower bound of-
exp(C(hp
+dP)3),
but the details are yet to appear. We will obtainour
general
resultby making
use of thefollowing corollary
of atheorem of M.
Mignotte
and M. Waldschmidt[10]:
Let a E C,
a = 0, log a -:é 0
with balgebraic
but irrational. There areabsolute constants
Co>
0 and to > 0 such thatfor
non-zeroP(x), Q(x)
EZ[x]
withwe have
Earlier work in this direction would have
sufficed,
except that itapplied only
whendeg
P +deg Q
was below some fixed bound.This work was conceived and
mainly
carried out at a conference in transcendencetheory
atCambridge University.
We would like tothank the
Department
of PureMathematics, Cambridge University,
but
especially
A.Baker,
J. Coates and D.W. Masser for theirhospi- tality.
The author alsogratefully
thanks D.W. Masser forhelpful
conlments on a
previous
version of this paper.1. Statement of Results
The first result
applies
to the situation of Gelfond and Feldman:THEOR FM 1. Let a E C be
algebraic,
a gé0, log
a 7é 0 and(3
E C be cubic irrational. Then there is apositive, effectively computable
con-stant C such that
for
every non-zeropolynomial Q(x, y)
EZ[x, y],
We use an asertisk to denote the maximum of 1 and the function value. Theorem 1 follows from the
proof
of the moregeneral
result:THEOREM 2. Let
a E C, a:,é 0, log a 0.
Let(3 E C be
a cubic irra- tional. Then there is apositive, effectively computable
constant D suchthat
for
allrelatively prime polynomials P(x, y), Q(x,
y,z)
EZ[x,
y,z],
we have that
where
Theorem 2
actually
holds in a somewhat stronger forminvolving
thepartial degrees
of P andQ.
Note 1.
Any
two of the three numbers a,a , a
could appear in the left handpolynomial
in Theorem 2 aslong
as thecorresponding polynomials P, Q
remainrelatively prime.
Note 2. If
Q(a, a13, a(32)
=0,
then we have asgood
a measure ofalgebraic independence
for any two of these numbers which areindependent
as Gelfond and Feldman did foralo
anda132
when a isalgebraic.
Thus we can conclude that when a is notalgebraic,
but isvery well
approximated by algebraic numbers,
then the three numbersa,
alo, a92
arealgebraically independent.
Anexample
of such an a iswhere ... stands for the intermediate consecutive
integers.
For whena is
irrational,
butQ(a, a{3, a(32)
=0,
then there is aneff ectively computable
C such that for rationalapproximations plq
to a, we haveNote 3. If
only
two of the numbers a,alo
anda a2
are involved in P andQ,
we can eliminate any common occurrence and then use the result ofMignotte
and Waldschmidt toproduce
our conclusion. Ifonly
one of the numbers isinvolved, forming
the resultantgives
theconclusion
immediately.
Note 4. It is
possible
to deduce a non-trivial lower bound forarbitrary pairs
ofrelatively prime polynomials P(x,
y,z), Q(x,
y,z)
inZ[x,
y,z]:
If z occurs in P andQ,
then formR,
the resultanteliminating
z between the smallestprime
factorsP,, d
of P andQ,
respectively. Apply
Theorem 2 to R andQI, using
Lemma 5 below.(R
and
QI
arerelatively prime by
the standardrepresentation
of R as apolynomial
linear combination ofPl
andQl.)
Finally,
consider theanalytic
curvewhere a’ denotes as usual
exp(z log a),
for some fixedlog
a. It follows from Theorem 2 that the coordinates exhibit nosurprising algebraic degeneracy.
Since any one of the coordinates may even bean
algebraic
number y(e.g.
when t =log y/log a),
we see that at timesthe
analytic
curve intersectsalgebraic
surfaces overQ,
thealgebraic
closure of
Q.
Butby
Remark4,
it does not even come close(in
anappropriate sense)
toalgebraic
curves overÕ,
except of course at t = O.What we
actually
will prove is thefollowing
more exactresult,
whereCo
still denotes the constant in the result ofMignotte
andWaldschmidt:
THEOREM 3. Let
a E C, a:;é 0, log a:;é O. Let f3 E C be
a cubicirrational. Then
for
agiven CI
>Co,
there is aneffectively computable
constant
C, depending only
onCh f3
andlog
a, such thatif
there aretwo
relatively prime polynomials P(x, y), Q(x,
y,z)
EZ[x,
y,z]
satis-fying
where
then there is a
polynomial U(x)
EZ[x]
withU(a) -:;é
0 andThe
proof
of thecorresponding inequality
forao
is obtained from thefollowing proof merely by interchanging a
andala everywhere
below. For
large enough C,
in terms ofCo,
theinequalities (2)
for aand
alo
are contradictedby
the result ofMignotte
andWaldschmidt,
thus
establishing
Theorem 2. Theorem 1 follows onletting
P be theminimal
polynomial
f or a over Zand,
ifdegyQ deg,Q, interchang- ing a la
anda02
in the argument.2. Proof of Theorem 3
We assume that Theorem 3 is false for
large enough
C.Step
0.Preliminary
Remarks. Without loss ofgenerality
we mayassume that both P and
Q
are irreducible over Z(see
Lemma2). By
our denial of
(2),
we havedegyP
> 0 and a non-trivial lower bound onthe
leading
coefficientp(a)
ofP(a, y):
Since there are at most
degyP
distinct roots, we are forced to admitthat a root
el
ofP (a, y)
= 0 nearesta o
satisfiesMoreover since
el -a" formally
dividessee
easily
thatThe
polynomial Q(a, el, z)
is notidentically
zero since P andQ
arerelatively prime.
If z did not occur after the substitution ofel
fora s,
then we would
simply
form the resultant(see
Lemma 5below)
ofP(a, y)
andQ(a,
y,z)
with respect to y to fulfill(2). Similarly
we evenobtain a lower bound on the
leading
coefHciëntq(a, el)
of z inQ(a, el, z). Arguing
as before forP,
we see that a roote2
ofQ(a, el, z)
= 0 nearest toa/32
satisfiesIn the
proof,
the Gothic lower case letters1,
n will be used to denotetriples
ofintegers given by
thecorresponding
Greekletters,
and absolute valuesigns
will denote the sup norm.E.g.
l=(Ào, À
1,À 2)
EZ(3)
andItl
= maxlÀ; 1.
The coordinates will be non-nega-tive
except
in Lemma 5. In addition weset b = (1, f3, (32),
l. b =ÀO+IklP + À2f32 and similarly
for n . b. We also letBi = (1 . b)b 2 log
a,where b is the
positive leading
coefficient of the minimalpolynomial
for
jg
over Z. The letters CI, C2, c3, ... will denotepositive
constantsdepending only
onf3
andlog
a. Forsimplicity
of notation in theproof,
we will assume that alldegrees
in theproof
are at least 1 andall
heights
at least 2.that there is a non-zero
polynomial
The
partial degrees
of p will be referred to asdegau
anddegeu.
Thefact that u
might
berepresented by
several such p will present noproblem.
Let
It is easy to
verify
that when C islarge enough,
thenWe note for use later in
Steps
4 and 5 that(i) since,
from the definition ofNo
and ofLN,
we know that
(ii)
sincewe know that
we know that
Step
1. We show that there existwithout a common nonconstant factor such that
and such that when we set the function
satisfies
for all
0 p DN and Irl LN/4.
For aftermultiplying
all theseexpressions by b 21N (a 1++2)e3NLN,
we obtain a linearhomogeneous system
of at most3N3/64 equations
inN3
unknownshaving
coefficients
polynomials
over Z in a,ap, a 02.
Weapply
Lemma 1below to solve them
formally,
i.e. with x, y, z substituted fora, a@, ’a .
We obtain the desired0,,(x,y,z)
ondividing
the formal solutions inZ[x,
y,z] by
theirgreatest
commondivisor, noting
thatNLN-24DNIogN,
andkeeping
in mind Gelfond’s result on theheight
of a factor(Lemma
2below).
Step
2. Anapplication
of theCauchy integral
formula on thecircle,
say,
Izi
=N3/2,
which is standard in transcendenceproofs (see,
e.g.[7,
pp.
157-158])
shows that for all0p DN and 111 LN log N,
Step
3. A result of R.Tijdeman’s (Lemma
4below)
shows that eitheror else
We now consider the two cases
separately.
Case
(i).
When wesubstitute §i
foraO
ande2
fora132
inwe obtain an element of
Z[a, bf3, el, e2]
which is non-zeroby
in-equalities (3), (4), (5).
Ontaking
the relative norm toZ [a, el, e2]
we geta non-zero q there with
and
Either the
leadirig
coefficientw(a, Çl)
ofq(a, el, z)
or elseq(a, el, ç2)/w(a, el)
has absolute value at mostWhen
w(a, el)
islarger,
we see from Lemma 5 and the fact thatQ(a, el, e2)
= 0 that the semi-resultantr(a, el)
EZ[a, el]
ofq(a, el, z)
and
Q(a, el, z)
satisfies theinequalities
which are also satisfied
by w(a, el)
when it is smaller.We
finally
take the resultant ofr(a, y)
orw(a, y)
andP(a, y)
to obtain a non-zero elements(a)
EZ[a] with
Case
(ii). By
our denial of(2),
we can assume a to be transcendental.Since the
0,,(x,
y,z)
are without a commonfactor,
not all the4>n(a, el, z)
are zero. It is conceivable that theleading
coefficient ofsome non-zero
4>n(a, el, z)
has absolute value at mostexp(- N3 log N/389)
and thus itself satisfies(8).
When the
leading
coefficients are at leastexp(- N3 log N/389),
wewill be interested in the monic
polynomials 1Jtn(z)
obtained from the4>n(a, el, z)
ondividing by
theleading
coefficient. Infact,
weapply
Lemma 5 to alter the definition of
FN(z)
somewhat. Forthis,
letQi(z)
be the minimal monic
polynomial
fore2
overQ(a, el).
Let a-, be the maximalnon-negative integer
such that for all n we can writefor some
polynomial R"(z),
whose coefficientsnaturally
lie inO(a, el).
If ai =
0,
then we obtain a non-zeror(a, Çl)
EZ[a, Çl]’ satisfying (8),
from Lemma 5
by forming
the semi-resultant ofQ(a, el, z)
and some4>n(a, el, z)
not divisibleby Ql(z).
To check the bound on the absolutevalue,
evaluateQl(z)
andIP,,(z)
at z =e2.
If (ri
> 0,
we define a new functionGN(z) by replacing cp(n)
in thedefinition of
FN(z)
withR,,(a 02 ).
Then for0 s p DN
and)1[
LN log
N theexpressions
are
polynomials,
saySp,N
=Y,aja°1,
ofdegree
at mo stc 1 NLN
with coefficients from0(jg,
a,el, a o2) having height
at mostexp(Cl6DN log N).
Recall that we solved for
0(x,
y,z) formally
inStep 1,
onreplacing
a
by
x,a,’ by
y anda ,2 by
z in theequation
forFN(z) corresponding
to
(10).
Therefore when weresubstitute el
forap
and z fora ,2
in theoriginal equations,
theexpressions
remain zero even after divisionby
Qi(z)"
and substitution anew ofa ,2
for z. ThusS,,N(el)
=0,
andconsequently
We now
apply
the Hermiteinterpolation
formula to find that forwhere m = min
IB, - Bpf; If l, lf’l LN. Using
standard estimates(see [4,
lot,
p.
69]
fordetails),
we see thatApplying
theCauchy integral
formula onrzr
=N3/2
shows that for allNow from the definition of o-1, we find that some
Rn(a")
is non-zero.However there are still two
possibilities
forGN,
as inStep
3 forFN.
If
GN
falls into Case(i),
then for the relevantPo, (0,
we cansubstitute
el
foral3
and z fora132
in(7)
andand form the semi-resultant of the
polynomial q(«, el, z)
obtained from(7)
andQ(«, el, z)
to obtain a non-zeror(a, ii)
EZ[a, el].
If theleading
coefficient w ofq(a, el, z)
has absolute value at leastwe consider
Ql(z)
and the monicpolynomial q(a, el, z)/w
evaluated atz =
e2
to see thatr(a, el)
satisfies(8). If,
on the otherhand, w is
lessthan the bound
cited,
it is less than the bound in(8).
Recall that(11)
was obtained
formally
from(7)
onreplacing
some of thea,’ (those coming
fromcp(n»
withel
anda 82
with z,dividing by Ql(Z)UI,
andthen
replacing
theremaining alo
withel
and z witha ,2 . Actually
the firststep
may as well have been toreplace
allala
withel.
NowQi(z)
is monic. So w is obtained from some coefficient ofa 62
in(7) by
replacing a a
withel,
and hence w EZ [a, el], satisfying
theremaining
inequalities
in(8).
At any rate, we have seen that ifGN
falls into Case(i),
we can find anr(a, el) satisfying
theinequalities (8).
If
GN
falls into Case(ii),
then wesimply
form the semiresultant ofQ(a, el, z)
and of a0,,(a, el, z)
for which (Tl is the exact power ofdivisibility by Ql(Z).
We then useQl(Z)
and(the
monicpolynomial
associated
with)
thecorresponding Rn(z)
evaluated ata R2
to show that the semi-resultant satisfies(8),
from ourprevious
agreement on the lower bound for the absolute value of theleading
coefficient ofcIJn( a, el, z)
and thus ofRn(z).
Thus we have shown that even in Case
(ii)
we can obtain anon-zero
r(«, el) satisfying (8). Taking
the resultant withP(«, y) gives
us an
s(a)
EZ[al satisfying (9) just
as in Case(i).
Step
4. There is a factortN (a)
=u(«)vN
ofs (a)
withu (a)
EZ[a]
irreducible over
Z,
vN ?1, satisfying
To
verify this,
refer to Lemma 3 below and theinequalities following
the definition of
LN
andDN. By
the same token we canapply
Lemma5 to tN and tN,I,
NO S
NNi
to see that theunderlying
irreduciblepolynomial u (a )
is the same for N and N + 1 and hence for allN, NoNN1.
Now since we see that
as desired.
By
the definition ofDN,
and
by inequality (i)
after the definition ofLN
andDN,
we see thatTherefore we can conclude from Lemma 2 below that
3. Lemmas used in the
proof
LEMMA 1: Let R and S be
positive integers
with 16R - S. Let aij EZ[x,
y,z) satisfy
where H %-- 1. Then the system
of equations
has a non -trivial solution CPh ..., CPs E
Z[x,
y,z]
withFor a
proof,
see[7,
Lemma2,
p.1351
or, for the fund-amental onevariable case,
[ 11,
Lemma3,
p.149]
or[9].
LEMMA 3 :
Suppose w OE C and P(x) G Z [x] , P(x) # 0, satisfy )P(w)j
e-Ad(h+d)
where k >3,
d >deg P, eh >
ht P. Then there is afactor Q(x)
of P(x)
which is a powerof
an irreduciblepolynomial
inZ[x]
such thatFor a
proof,
see[7,
LemmaVI,
p.147].
where in the
definition of Bo,
the coordinatesof
n may benegative,
butnot all zero, and
This is a
special
case of whatTijdeman proved
in[12,
Theorem3,
pp.87-88].
the semi-resultant r of P and
Q
is definedby
where ’ means that the
product
is taken over all(i, j) F- M
={(i, j) 1
ti =Uj}. Clearly
r is non-zero.LEMMA 5
(Choodnovsky):
LetP(x)
=poxm
+ ... + p,,,Q(x) = qox"
+ ... + qn EC[x]. (i)
Their semi-resultant r can be written as apolynomial
over Z in the pi and qi with(ii)
Let 0 EC,
and letPi(x), Ql(x)
EC[x]
bemonic, relatively prime,
and
satisfy
Then when
PI(x)
andQI(x)
divideP(x)
andQ(x), respectively,
where k is an absolute constant.
For a
proof,
see[6,
Lemma4.12]
or[3].
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210 (1975) 1-26.
[2] W.D. BROWNAWELL: Pairs of polynomials small at a number to certain algebraic
powers, Séminaire DELANGE PISOT POITOU, 17e année (1975/76) n° 11.
[3] W.D. BROWNAWELL: Some remarks on semi-resultants, pp. 205-210, in Advances in Transcendence Theory, A. Baker and D.W. Masser, eds., Academic Press, 1977.
[4] W.D. BROWNAWELL and M. WALDSCHMIDT: The algebraic independence of
certain numbers to algebraic powers. Acta Arith. 32 (1977), 63-71.
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[9] K. MAHLER: An application of Jensen’s formula to polynomials, Mathematika 7
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[10] M. MIGNOTTE and M. WALDSCHMIDT: Linear forms in logarithms and Schneider’s Method. Math. Ann. 231 (1978) 241-267.
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(Oblatum 15-IX-1977 & 73-11-1978) Department of Mathematics Pennsylvania State University University Park, Pa. 16802 U.S.A.