Integer solutions to decomposable form inequalities

This paper obtains a result on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X1,...,Xm) be a non-degenerate decomposable form with coefficients in k. We prove that, for every finite set of places S of k containing the archimedean places of k, for each real number λ<1m-1 and for each constant c>0, the inequality(1)0<∏υ∈S∥F(x1,...,xm)∥υ⩽cHSλ(x1,...,xm)in(x1,...,xm)∈OSm.has only finitely many OS*-non-proportional solutions.