Decomposable form inequalities

Abstract

We consider Diophantine inequalities of the kind |f(x)| \le m, where F(X) \in Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m\ge 1. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every n'-dimensional subspace S\subseteq R^n defined over Q, the corresponding n'-dimensional volume for F restricted to S is also finite. We show that the number of integral solutions x \in Z^n to our inequality above is finite for all m if and only if the form F is of finite type. When F is of finite type, we show that the number of integral solutions is estimated asymptotically as m\to \infty by the total number of integral solutions is estimated asymptotically as m\to \infty by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case n=2. Further, we prove a conjecture of W. M. Schmidt, showing that for F of finite type the number of integral solutions is bounded above by c(n,d)m^(n/d), where c(n,d) is an effectively computable constant depending only on n and d.