Determining the small solutions to S-unit equations

In this paper we generalize the method of Wildanger for nding small solutions to unit equations to the case of S-unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes. The main computational problem when solving a diophantine equation is usually the location of the \small" solutions. In this paper we assume we are given the generators of two nitely generated multiplicative subgroups of some number eld, K. In what follows we shall denote these subgroups by G1 and G2. We also assume that we are given two xed algebraic numbers 1; 2 2 K . In [4] the author gave a practical algorithmic solution to the determination of all the solutions to the equation 1 1 + 2 2 + 1 = 0 with ( 1; 2) 2 G1 G2: (1) That there are nitely many solutions to such an equation follows from work of Siegel. An e ective proof of the niteness of the number of solutions was rst given by Gy}ory, [3], using Baker's method of linear forms in logarithms. Using an adaption of Gy}ory's method combined with the reduction techniques of de Weger, [9], one can reduce the solution of (1) to the determination of the \small" solutions. The technique used in [4] to determine such solutions was a sieving technique which lent itself to implementation on a parallel computer or a network of workstations. For further discussion of this sieving technique see [5]. Recently Wildanger, [10], has given a much more e cient technique of determining the small solutions in the case where G1 = G2 = O K . In this paper we extend Wildanger's method to the general case. The main problem that one encounters is the presence of nite places in the support of the two groups. Wildanger makes use of the algorithm of Fincke-Pohst, [2]. We try to avoid the use of this algorithm for as long as possible. This is because we feel that applying Fincke-Pohst to lattices generated by real vectors with very large coe cients held to very high precision can lead to oating point errors. This is due to rounding errors in the algorithm for Cholesky decomposition and in the LLL algorithm itself. Indeed rounding errors introduced in the oating point version of the LLL algorithm can lead to the production of a basis which is not even LLL reduced. Below we make use of the LLL algorithm on lattices generated by vectors with integer entries. We can therefore make use of the integer version of the LLL algorithm due to de Weger, [8], which does not su er from numerical instability. We only apply the algorithm of Fincke-Pohst and the oating point version of LLL when we have reduced considerably the precision needed in the calculations. 1 1. Notation We shall let S1 and S2 denote the set of primes (places), both nite and in nite, in the support of the groups G1 and G2 respectively. In other words Si = fp 2MK : j jp 6= 1 for some 2 Gig = Supp(Gi): We let ti denote the rank of the group Gi. We suppose that Gi has generators of in nite order given by 1;i; : : : ; ti;i. We can then write