We consider the equation
a1x1 + ··· + anxn = 1, (1.1)
where the coefficients a1, …, an
are non-zero elements of a field K, to be solved for
(x1, …, xn) in a finitely generated
multiplicative subgroup G of (K*)n.
Evertse [2] and
van der Poorten and Schlickewei [6] showed, independently, for the most
important case in which K is a number field and G is a group of
S-units in K, that (1.1) has
only finitely many solutions (x1, …, xn)
for which no subsum in (1.1) vanishes. Such
solutions are called non-degenerate.
In 1990, Schlickewei [7] was the first to
obtain an explicit upper bound for the number of non-degenerate solutions of (1.1) when
G is a group of S-units of K. His
result went through various successive improvements, and the best result to date is
the bound (235n2)n3s
by Evertse [3, Theorem 3], where s
is the cardinality of S. Very
recently, Evertse, Schlickewei and Schmidt [4, 5]
obtained a remarkable result. They
showed, again for K a number field, that the number of non-degenerate solutions of
(1.1) with (x1, …, xn) in a
finitely generated subgroup of (K*)n of rank r is at most
c(n)r+2, with c(n) =
exp((6n)4n). The importance of this
result lies with its uniformity with respect to the rank r of the group
and its independence of the field K. The
proofs of the above-mentioned results are all quite difficult and depend on deep
tools from Schmidt's Subspace Theorem and diophantine approximation. In this
paper we consider equation (1.1) in the rational function field
K = k(t) where k is
an algebraically closed field of characteristic 0. Of course, results of this type will
follow rather easily from the above-mentioned results by a specialization argument.
However, our object is to show that our approach to the function field case is quite
elementary and certainly very different, being a simple and direct consequence of
the powerful abc-theorem.
Before stating our result, we first define proportional solutions of (1.1). We
say that (x1, …, xn) and
(x′1, …, x′n) are
proportional if
xi, /, x′i ∈ k,
1 [les ] i [les ] n. This determines equivalence classes of solutions.