Upper bounds for the degrees of decomposable forms of given discriminant
1. Introduction. In our paper  a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author  to decomposable forms in n ≥ 2 variables. The bound obtained in  depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish an improvement of the bound of  which is already best possible for every n ≥ 2. Moreover, all the squarefree decomposable forms in n variables over ℤ will be determined for which our bound cannot be further sharpened. In the proof we shall use some results and arguments of  and  and two theorems of Heller  on linear systems with integral valued solutions.