ON THE PROBLEM OF INTEGER SOLUTIONS TO DECOMPOSABLE FORM INEQUALITIES

This paper proves a conjecture proposed by Chen and Ru in [1] 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 show that for every finite set of places S of k containing the archimedean places of k, for each real number λ < 1 and each constant c > 0, the inequality [Formula: see text] has only finitely many [Formula: see text]-non-proportional solutions, where HS(x1,…,xm) = Πυ∈S max 1≤i≤m ||xi||υ is the S-height.