Upper Bounds for the Number of Integral Points on Quadratic Curves and Surfaces

Abstract

We are interested in investigating the number of integral points on quadrics.

First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic.

Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.