On the number of solutions of norm form equations

Abstract

In 1985 Evertse and Gyory [5] gave explicit upper bounds for the number of solutions of norm form equations of the form (1.1) under the hypotheses that (i) x m n 0, α 1 = 1, α 2 , ... ,α m-1 are Q-linearly independent and has degree at least 3 overQ( α 1 ..., α m-1 ), or that (ii) the degree of i is at least 3 over Q(α 1 , ..., α i-1 ) for i = 2, m. Later Győry [9], Evertse [3] and Evertse and Gy}ory [6] derived general upper bounds for arbitrary norm form equations which include the case (ii), but not the case (i). In the present paper we considerably improve the bounds of [5], and we give a further improvement which is valid for all but at most finitely many possible values of the constant term b of the equation. Our bound obtained under the assumption (ii) is better for almost all b than the general bounds of [9], [3] and [6].