The problem of finding the number of solutions of an equation as above is an old one, see [Maľsev, 1962; Maľsev, 1964; Maľsev, 1974; Ozeki, 1973], but most of the positive results are upper bounds on arguments of volume [Baranovskĭı, Ryshkov and Shushbaev, 1983] or asymptotic limits obtained ∗During the preparation of this paper, the author was partially supported by a certain CNPq grant 20.1553/82-MA.

A geometric estimate of the number of representations of a real number by a positive quadratic form and an estimate of the remainder of the multidimensional zeta-function

Proc. Steklov Inst. Math. 158:

1983

Problems in low dimensional manifold theory

Proc. Symp. Pure Math.,

1978

The formulae for the number of representations of numbers by positive quadratic forms, Current Problems in Anallitic Number Theory, Izdat

Nauka i Tehnika. Proc. Summer School,

1974

A Course in Arithmetic, Vol. 7 of Grad

Texts in Math, Springer-Verlag, New York. Transl. of [Serre,