Auxiliary Equations

Abstract

Index form equations can often be reduced to simpler types of diophantine equations. This is the case e.g., when the index form factorizes. As we shall see in the following chapters, various types of Thue equations play an essential role in the resolution of index form equations [Ga96b]. We summarize the methods for the resolution of these equations in this chapter. We shall consider Thue equations (Section 3.1), inhomogeneous Thue equations (Section 3.2) and relative Thue equations (Section 3.3). The algorithms for solving them are based uniformly on the tools of the preceeding chapter. In this chapter we also include an algorithm for solving certain types of norm form equations (Section 3.4), the type of the equation and the ideas for solving it being very close to what we use for the various types of Thue equations.