Volumes and diophantine inequalities associated with decomposable forms

AbstractFor homogeneous decomposable forms F(X) in n variables with real coefficients, we consider the associated volume of all real solutions x∈Rn to the inequality |F(x)|⩽1. We relate this to the number of integral solutions z∈Zn 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