Chapter VIII - The Superelliptic Equation

Summary

Hitherto this book has effectively resolved the problem of determining the complete set of integral solutions to certain general families of equations. The fundamental inequality has formed the crux of the argument in each analysis, and it has led to the solution in Chapter II of the Thue equation, in Chapter III of the hyperelliptic equation, and in Chapter IV of equations of genera 0 and 1. In Chapter VII we succeeded in dealing with the Thue and hyperelliptic equations over fields of positive characteristic, and it was the appropriate extension of the fundamental inequality to such fields which again provided the crucial step. In the case of positive characteristic it is possible for the heights of the integral solutions to be unbounded, but this cannot occur over fields of characteristic 0, and for that case we determined explicit bounds for each of the various families solved. The fundamental inequality contributed the essence to each of those proofs also, and it is the purpose of this concluding chapter to illustrate a further range of applications for the inequality by studying briefly the superelliptic equation. Here the inequality is employed in a rather different fashion from previously, and this new approach will in fact lead to explicit bounds on the heights of all the solutions, not just those integral. Explicit bounds for non-integral solutions have only been obtained by Schmidt [36] in the case of certain Thue equations, and stronger bounds may be deduced from our methods as below.