An asymptotic expression for the number of solutions of a general class of Diophantine equations

Consider a closed, strictly convex body C defined by/(xi, • ■ ■ , x") =P.If f(xi, ■ ■ • , x") is a homogeneous function, it is easily verified that N, the number of solutions of /(xi, ■ ■ ■ , x")=P in integers, satisfies the inequality cRn~1>N.The object of this paper is to show that this inequality may be replaced by cP(n_1)n/(n+1)> N.This result will be derived from the following theorem.