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.