Lattice points below algebraic curves

Lattice points below Algebraic Curves. A generalization of the classical circle problem is treated. An asymptotic formula For the number of lattice points in a region whose boundary is an algebraic curve is obtained. This gives a mean value Formula for the number of representations of the positive integers in the form n = g(x,y), where g is a polynomial with coefficients greater than or equal to 0 and leading term a(d0)x(d) + a(0d)y(d). The case g(x,y) = p(1)(x) + p(2)(y) was considered in KUBA and NOWAK [4], and KUBA [5]. The discrete Hardy-Littlewood method is used along with Rouche's theorem.