and 
by
and Abstract. We give two measures of simultaneous approximation
by algebraic numbers, the first one for the triple 
)
and the second one for 
).
We deduce from these measures two transcendence results which had been
proved in the early 70's by W.D. Brownawell and the author.
and 
are algebraically
independent. At the same time, he proved that one at least of the three
numbers 
is transcendental
(see[G] Chap. III, ).
At the end of his book [S] on transcendental numbers, Th. Schneider
suggested that one at least of the two numbers 
is transcendental; this was the last of a list of eight problems, and the
first to be solved, in 1973, by W.D. Brownawell [B] and M.Waldschmidt [W
1], independently and simultaneously. For this result they shared the Distinguished
Award of the Hardy-Ramanujan Society in 1986. Another consequence of their
main result is that one at least of the two following statements holds
true:
(i) The numbers e and 
are algebraically independent
(ii)The number 
is
transcendental
Our goal is to shed a new light on these results. It is hoped that our approach will yield further progress towards a solution of the following open problems:
(?) Two at least of the three numbers 
are algebraically independent.
(?) Two at least of the three numbers 
are algebraically independent.
Further conjectures are as follows:
(?) Each of the numbers 
is transcendental
(?) The numbers e and 
are algebraically independent.
We conclude this note by showing how stronger statements are consequences of Schanuel's conjecture.
be a complex algebraic
number. The minimal polynomial of 
over Z Z is the unique polynomial
which vanishes at the point 
,
is irreducible in the factorial ring 
and has leading coefficient 
.
The integer 
is the degree
of 
, denoted by 
.
The usual height 
of 
is defined by
It will be convenient to use also the so-called Mahler's measure
of 
, which can be defined
in three equivalent ways. The first one is
For the second one, let 
denote the complex roots of f, so that
Then, according to Jensen's formula, we have
For the third one, let K be a number field (that is a subfield
of C| which is a Q/ -vector space of finite dimension [K:Q/ ]) containing 
and let 
be the set of
(normalized) absolute values of K. Then
where 
is the
completion of K for the absolute value v and 
the topological closure of 
in 
and 
the local degree.
Mahler's measure is related to the usual height by
From this point of view it does not make too much difference to
use H or M, but one should be careful that d denotes
the exact degree of 
, not
an upper bound. We shall deal below with algebraic numbers of degree d
bounded by some parameter D.
Definition. For an algebraic number 
of degree d and Mahler's measure 
,
we define the absolute logarithmic height 
by
2.1. Simultaneous Approximation to 
and 
Theorem 1. There exists a positive absolute constant 
such that, if 
are algebraic
numbers in a field of degree D, then
where 
2.2. Simultaneous Approximation to 
and 
Theorem 2. There exists a positive absolute constant 
such that, if 
are algebraic
numbers in a field of degree D, then
where 
The following result is Théoréme 3.2 of [R-W 1]; see also Theorem 1.1 of [R-W 2].
Theorem 3. Let 
be a complex number. The two following conditions are equivalent:
(i) the number 
is transcendental.
(ii) For any real number 
,
there are infinitely many integers 
for which there exists an algebraic number 
of degree d and absolute logarithmic height 
which satisfies
Notice that the proof of 
is an easy consequence of Liouville's inequality.
3.2. Application to 
and 
Corollary to Theorem 1. One at least of the two numbers 
is transcendental.
Proof of the corollary. Assume that the two numbers 
are algebraic, say 
and 
.
Then, according to Theorem 1, there exists a constant 
such that, for any algebraic number 
of degree 
and height 
with 
We now use Theorem 3 for 
with 
and derive a contradiction.
The proof of the following result is given in [R-W 1], Théoréme 3.1, as a consequence of Theorem 3 (see also [R-W 2] Corollary 1.2).
Corollary to Theorem 3. Let 
be complex numbers such that the field 
has tyranscendence degree 1 over Q. There exists a constant c > 0
such that, for any real number 
,
there are infinitely many integers D for which there exists a tuple 
of algebraic numbers satisfying
and
4.2. Application to 
and 
Corollary to Theorem 2. One at least of the two following
statements is true:
(i) The numbers e and 
are algebraiclly independent.
(ii) The number 
is transcendental.
Remark. This corollary can be stated in an equivalent way as follows:
For any non constant polynomial 
,
the complex number
is transcendental.
The idea behind this remark originates in [R].
Proof of the Corollary. Assume that the number 
is algebraic. Theorem 2 with 
shows that there exists a constant 
such that, for any pair 
of algebraic numbers, if we set
then
Therefore we dedeuce from the Corollary to Theorem 3 that the
field 
has transcendence
degree 2.
Schanuel's Conjecture. Let 
be Q-linearly independent complex numbers. Then, among the 2n numbers
at least n are algebraically independent.
Let us deduce from Schanuel's Conjecture the following statement (which is an open problem):
(?) The 7 numbers
are algebraically independent.
We shall use Schanuel's cinjecture twice. We start with the numbers 
and 
which are linearly
independent over Q because 
is irrational. Therefore, according to Schanuel's conjecture, three at
least of the numbers
are algebraically independent. This means that the three numbers 
and e are algebraically independent. Therefore the 8 numbers
are Q-linearly independent. Again, Schanuel's conjecture implies that 8 at least of the numbers
are algebraically independent, and this means that the 8 numbers
are algebraically independent.
ACKNOWLEDGEMENTS.
This research was performed in March/April 1998 while the author visited the Tata Institute of Fundamental Research under a cooperation program from the Indo-French Center IFCEPAR/CEFIPRA Research project 1601-2 << Geometry >>, The author wishes also to thank Professor K.Ramachandra for his invitation to the National Institute of Advanced Studies at Bangalore.
E-mail: miw@math.jussieu.fr
Received 28th March 1998
Accepted 30th April 1998.
Hardy-Ramanujan Journal
Vol.21 (1998) 35-36