Acta Mathematica Academiae Scientiarum Hungaricae 69. (1995)

1995 / 1-2. szám - Győry K.: On a problem of A. M. Odlyzko on algebraic units of bounded degree

2 К GYŐRY Theorem. We note that the proofs in [2] depend, among other things, on H. P. Schlickewei’s p-adic generalization (cf. [11]) of W. M. Schmidt’s quan­titative Subspace Theorem [12]. Our explicit bound concerning decomposable form equations has recently been improved by J. H. Evertse (private communication). Using his improve­ment, one can prove that (2) M(n) < exp{ 36n2n+5} . Finally, we mention that using Theorem 3 of the author [3], the existence of M{n) can be proved in the more general situation as well, when the ground ring Z is replaced by an arbitrary finitely generated and integrally closed integral domain over Z. 2. Proof of the Theorem Let К be an algebraic number field of degree к with ring of integers Од­­and unit group 0*K. Let F(xo,x\,... ,xq) (q ^ 1) be a decomposable form of degree t with coefficients in Од, i.e. a homogeneous polynomial which factorizes into linear factors over a finite extension G of K. Two solutions ж, x' of the decomposable form equation (3) F(x0,xi,...,xq) 6 0*K in i = (i0v,í5)éOJ1+1 are called proportional if x' = ex for some e E 0*K. Let d denote the degree of the normal closure of G over Q. To prove our Theorem, we need the following. Lemma. Suppose that t > 2q and that any 9 + 1 linear factors in the factorization of F are linearly independent. Then equation (3) has at most (4) (5 Uf"‘k‘ pairwise non-proportional solutions. Proof. This is an immediate consequence of Theorem 6 of the author [4] . Its proof involves, among other things, an estimate of Schlickewei [10] for the number of solutions of 5-unit equations. □ We note that a more general but qualitative version of the Lemma was proved in [3] over an arbitrary finitely generated integral domain over Z. Remark 1. On combining the above-mentioned result of Evertse with the proof of Theorem 3 of [3], our Lemma can be proved with the bound (5) (234t2)?3fc. Acta Mathematica Hungarica 69, 1995

Next