In this chapter we will continue with the task of describing the known Diophantine classes of the rings of W-integers of global fields. We will start with horizontal problems. The question which we will partially answer here is the following. Does the Diophantine class of a ring of W-integers change if we add to or remove from W finitely many primes? As we will see below, we are able to show in many cases that the class does not change. We conjecture that this is true for all rings of W-integers but are unable to prove this at the present time.

The main tool used so far to prove results of the type described in this section is the strong Hasse norm principle (see Theorem 32.9 of [79]). The ideas behind the construction of a Diophantine definition of integrality at finitely many primes presented below go back to the work of Ershov and of Penzin (see [65], [26]) and to the work of Julia Robinson on the arithmetic definability of rational integers in algebraic number fields (see [81] and [82]). Robinson used quadratic forms to carry out her construction. Later, Rumely generalized Robinson's methods in his paper on arithmetic definability over global fields (see [85]). In his paper Rumely used norm equations and the strong Hasse norm principle. Kim and Roush were the first to use this methodology for the purposes of showing the undecidability of some Diophantine problems over function fields (see [42]).