Chapter IV - Equations of Small Genus

Summary

INTRODUCTION

This chapter will be devoted to the construction of algorithms whereby may be determined the complete set of integral solutions of all equations of genera 0 and 1 over an arbitrary algebraic function field K. As usual [40] we shall assume that in the former case the curve associated with the equation possesses at least three infinite valuations. As for the Thue and hyperelliptic equations already solved, in each case we shall establish a simple criterion for the equation to possess an infinity of solutions in 0. In the next chapter a constructive examination of the algorithms will be used to establish explicit bounds on the heights of the solutions. The chief ingredient in those proofs is a direct recursive technique for determining the coefficients in a Puiseux series (see Lemma 5 and [10]). The bounds obtained are linear functions of the height of the original equation and thus are not exponential, as are the bounds established by Baker and Coates [8] in the case of algebraic numbers. This provides further evidence of the strength and power of our fundamental inequality which again plays the crucial role in the analysis. These results on the heights of solutions may be viewed in another way, as a complement to the celebrated theorem of Manin and Grauert, which had proved the analogue for function fields of Faltings' recent result. We recall that, as a consequence of this theorem, the heights of all solutions in K, not just those in 0, of any equation of genus 2 or more, are bounded.