Many books have been devoted to the theoretical study of diophantine equations, an observation which should come as no surprise given that the study of such equations dates back over two thousand years. In theoretical work one is interested in determining the structure of the solution set to some equation. Is the set finite or infinite? Can one give an effective procedure to determine all the solutions? Do the solutions form a group of some sort? How are the rational solutions distributed amongst the real solutions? The list of questions that one can ask is endless.

In this book we shall concentrate on algorithms and methods for writing down all the solutions to an equation (if there are finitely many) or for determining explicitly the structure of all of the solutions (if there are infinitely many). Despite the long and noble career of diophantine equations, there appear to be only two books solely devoted to the study of explicit methods for their solution, namely Mordell's Diophantine Equations [138] and de Weger's Algorithms For Diophantine Equations [208].

Mordell's book gives a variety of techniques for solving various diophantine equations. However, sometimes he deals just with special cases and sometimes with general cases. Mordell does not concentrate on algorithmic questions and hence some of his methods appear at first sight to be recipes which only apply to certain special cases.