Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms F(X) in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality F(x) 1. We relate this to the number of integral solutions to the Diophantine inequality F(z) m in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.