This chapter is devoted to the existential definability of bounds on the height of a number field element. We will consider two ways of bounding the height. The first method (due to Denef in [18]) relies on quadratic forms and is most effective over totally real fields. The second method relies on divisibility and thus depends on the choice of ring. The influence of divisibility depends on whether any primes are allowed to occur in the denominator of the divisors of elements of the ring. We will discuss this aspect of the matter in detail below. Denef was also the first to use the divisibility method for bounding height over the rings of integers in the context of existential definability (see [18] again). The present author extended Denef's divisibility method for use with bounds in rings of algebraic numbers. These kinds of bound were used in [99], [101], and [106] and in a slightly modified way in [73].

Real embeddings

In this section we show how to use quadratic forms to impose bounds on archimedean valuations when the corresponding completion of the number field is R. The following lemma and its corollary constitute the technical foundation of the method. They are taken from [18].