S -Unit Equations in Function Fields Via the abc -Theorem

Abstract

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 ( 2 35 n 2 ) n 3 s 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⩽i⩽n. This determines equivalence classes of solutions. 1991 Mathematics Subject Classification 11D72.