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

Acta Math. Hangar. 69 (1-2) (1995), 1-4. ON A PROBLEM OF A. M. ODLYZKO ON ALGEBRAIC UNITS OF BOUNDED DEGREE K. GYÖRY* (Debrecen), corresponding member of the Academy To Professor K. Tandori on his 7Oth birthday 1. Introduction For an algebraic number field К, denote by M(K) the maximal length m of a sequence (£j,... ,£m) in К such that — £j is a unit for all г, j with 1 = * < j = m. Here we may assume without loss of generality that £\ = 0, £2 = 1 and £3,... ,£m are units. This can be achieved by translation and by multiplication by a unit. By a theorem of H. W. Lenstra Jr. [5], the number field К is Euclidean for the norm provided that M(K) exceeds the square root of the discriminant of К in absolute value times a (number-geometric) coefficient which depends only on the signature of К. It was also proved in. [5] that M(K) ^ The above-quoted theorem of Lenstra was used by Lenstra [5], A. Leutbecher and J. Martinet [6], J.-F. Mestre [8] and others to give several hundred new examples of Euclidean number fields. For related results and further references, see e.g. [7], [1] and [9]. For given positive integer n, denote by M(n) the maximal number m of algebraic units £i,...,£m of degree ^ n over Q (which can lie in different number fields) such that £,• — £j is a unit for all distinct i,j with 1 ^ i,j й 5í m. In a letter in February 1985, A. M. Odlyzko proposed the following problem: What is the value of M(n) ? Clearly, M(n) ^ M(K) - 1 for all algebraic number fields К of degree n. In 1985, I was able to prove M(n) < 00 only. In the proof I needed the use of the Thue-Siegel-Roth-Schmidt method. Theorem. We have (1) M(n) < expexp{39n(n2"+1)!}. We shall reduce the problem to n decomposable form equations. Then an explicit upper bound of ours for the number of solutions of such equa­tions (cf. [2], [4] and the Lemma in Section 2) will be applied to prove the * Research supported in part by Grant 1641 from the Hungarian National Foundation for Scientific Research. 0236-5294/95/$ 4.00 (c) 1995 Akadémiai Kiadó, Budapest

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