We introduce some terminology. Let
be the multiplicative group
of
non-zero complex numbers. Let be a subgroup of
.
is said to be a torsion group if all its elements have finite
order, that is, are roots of unity. In that case we say that has
rank 0.
More generally, is said to be
*of finite rank* if there are
with the following property:
for every
there exist integers
and a positive integer such
that
.
If is not a torsion group then
the smallest for which such
exist is called the rank of .

For instance, the group

root of unity |

has rank . More generally, any subgroup of containing has rank .

First let be non-zero rational numbers and let
be the multiplicative
group
generated by the prime numbers
. In 1933,
Mahler [17] showed that the equation

in
(1)

has only finitely many solutions.
In 1960, Lang [13] showed that
for any
and any subgroup of
of finite rank,
the number of solutions of equation (1) is finite.

For subgroups of there are reasonably efficient algorithms to determine all solutions of (1). For instance, consider the equation

Our concern is to give uniform upper bounds for the number of solutions
of
infinite classes of equations of the shape (1),
depending on as few parameters as possible.
In 1984, the author [4]
showed that in Mahler's case,
that is, with
and with the group generated by prime
numbers
,
equation (1) has at most
solutions.
This bound is independent of the primes
and of the
coefficients .
Building further on work of Schlickewei, in 1996 Beukers and Schlickewei
[1] proved the following general result:
*For any subgroup of
of finite rank , and any
,
equation (1) has at most
solutions.*

We mention that in 1988, Erds, Stewart and Tijdeman [3]
proved a
result in the other direction:
*Let be non-zero rational numbers.
Then for every
and every sufficiently large , there
is a subgroup of
of rank
such that (1) has at least
solutions.*

We now turn to equations in
variables, namely

in
,
(2)

where is a subgroup of
of finite rank, and
.
Assume that is not finite.
A solution of equation (2) is called *non-degenerate*,
if each subsum of the left-hand side is non-zero, i.e.,

It follows from work of van der Poorten and Schlickewei, the author, and
Laurent
from the 1980's that (2) has only finitely many solutions. The
major
tool in the proof of this result is W.M. Schmidt's *Subspace
Theorem*
(see next section).
In 1990, Schlickewei [23] was the first to give an explicit
upper bound for the number of non-degenerate solutions of (2),
but only in the special case that is contained in an algebraic
number field.
Schlickewei's bound depended, apart from the number of variables and
the
rank of , on several other parameters and when his work appeared,
it was an open problem to deduce a uniform upper bound depending only on
and the rank of .
After several intermediate results,
Schlickewei, Schmidt and the author [8], see also the survey
paper
[6]
succeeded in proving the following theorem:
**Theorem 1.**
*Let be a subgroup of
of finite rank , and let
.
Then equation (2) has at most
non-degenerate solutions
*

The basic tool was a new quantitative version of Schmidt's Subspace Theorem, obtained by Schlickewei and the author [7] or the survey paper [6]). The upper bound in Theorem 1 is probably far from best possible, but one can show that the theorem does not remain valid if the upper bound is replaced by a bound independent of or .

We mention that recently, Moree, Stewart, Tijdeman
and the author [5] and independently Granville (unpublished)
proved the following generalization of the result of Erds, Stewart
and Tijdeman
mentioned above:
**Theorem 2.**
*Let
be non-zero rationals. Then for every
and every sufficiently large there is a subgroup of
of
rank
such that has at least
non-degenerate solutions.
*

The proof is not based on Diophantine approximation
but uses instead some analytic number theory.