Volumes and diophantine inequalities associated with decomposable forms

Abstract

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