In this chapter we apply the method to solve S-unit equations, developed in the previous chapter, to solve a class of diophantine equations called triangularly connected decomposable form (TCDF) equations. These were first studied by Gyory, see [89], [93], [92] and [91], who gave effective upper bounds on their solutions. A practical algorithm was given in [180], and it is this method which we shall explain here. We shall see that both Thue and Thue–Mahler equations are examples of TCDF equations. Other equations also fall into this category, for instance discriminant and index form equations. While we shall develop the following for forms with integer coefficients and variables, almost all of what we shall say goes over verbatim to when the coefficients of the form and the variables lie in some ring of integers of a number field. For instance this allows one to solve Thue and Thue–Mahler equations defined over a ring of integers [183]. However, there is often a muchbetter method, see for instance [211].

Triangularly connected linear forms

We first consider what it means for a set of linear forms to be called triangularly connected. We shall also show how one can easily determine whether or not a set of linear forms is triangularly connected or not.

Let ℒ denote a set of m linear forms in v variables with coefficients in the ring of integers of some number field K.