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S ‐Unit Equations in Function Fields Via the abc ‐Theorem
Author(s) -
Mueller Julia
Publication year - 2000
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460939900675x
Subject(s) - mathematics , combinatorics , rank (graph theory) , function field , multiplicative function , group (periodic table) , degenerate energy levels , field (mathematics) , cardinality (data modeling) , upper and lower bounds , multiplicative group , unit (ring theory) , algebraic number field , function (biology) , zero (linguistics) , discrete mathematics , pure mathematics , mathematical analysis , quantum mechanics , physics , mathematics education , evolutionary biology , biology , linguistics , philosophy , computer science , data mining
We consider the equation a 1 x 1 +…+ a n x n = 1, (1.1) where the coefficients a 1 ,…, a n are non‐zero elements of a field K , to be solved for ( x 1 ,…, x n ) 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 ( x 1 ,…, x n ) 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 35n 2 )n 3 sby 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 ( x 1 ,…, x n ) in a finitely generated subgroup of ( K * ) n of rank r is at most c ( n ) r 2 , with c ( n ) = exp((6 n ) 4 n ). 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 ( x 1 ,…, x n ) and ( x ′ 1 ,…, x ′ n ) are proportional if x i ,/, x ′ i ∈ k , 1⩽ i ⩽ n . This determines equivalence classes of solutions. 1991 Mathematics Subject Classification 11D72.