On the distribution of integer points on curves of genus zero

Let C A n be a geometrically irreducible aane curve of (geometric) genus 0 deened over Z such that C(Z) is an innnite set. We use elementary methods to describe the distribution of C(Z) in C(R) relative to the real topology. The study of integral points on aane curves has a long history. The deenitive niteness theorem, due to Siegel 5], says that a geometrically irreducible aane curve C has only nitely many integral points unless it has geometric genus 0 and at most 2 points at innnity. In those cases where Siegel's theorem allows the possibility of innnitely many integral points, there is an underlying (additive or multiplicative) group action which can be used to describe the distribution of the integral points. In this note we will prove two theorems illustrating this last statement. As a corollary, we will prove a conjecture of Rojas 3] concerning the distribution of integral points relative to the real topology. Little of the material in this note will be new to the \experts," but we hope that an elementary exposition will be a useful addition to the literature. Remark. A classical work of Runge 4] gives a necessary condition for a plane curve C : f(x; y) = 0 to possess innnitely many integral points in terms of the Puiseux expansion of the point(s) of C at innnity. Using similar methods, Ayad 1] extends Runge's work and gives a classiication for plane curves similar to the description given below. We will not make use of Puiseux expansions in this paper. We set the following notation, which will remain xed throughout this note: C A n , a geometrically irreducible aane curve of (geometric) genus 0 deened over Z. C P n , the Zariski closure of C in P n. C 1 = (C r C)(C), the set of point(s) \at innnity" on C. C(Z) the set of integral points on C.