Some applications of decomposable form equations to resultant equations

1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications of Theorems 1 and 2 are also presented to polynomials in R[X] assuming unit values at many given points in R (cf. Corollary 1) and to arithmetic progressions of given order, consisting of units of R (cf. Corollary 2). Further applications to irreducible polynomials will be given in a separate paper. Our Theorem 3 seems to be interesting in itself as well. It is deduced from some general results of Evertse and the author [3] on decomposable form equations. Since the proofs in [3] depend among other things on the Thue-Siegel-Roth-Schmidt method and its p-adic generalization