In this chapter we will discuss some bound equations specialized for function fields. These bounds will be used in the next chapter in our discussion of Diophantine classes of function fields. Some methods used below should be familiar to the reader from Chapter 5.

Height bounds

In this section we will consider how to obtain information about the height of a function, given information on the height of a polynomial evaluated at this function. We also compare the height of the coordinates of a field element with respect to a chosen basis and the height of the element itself. (The reader is reminded that the definition of the height of a function field element can be found in B.1.25.)

Lemma 9.1.1. Let K be a function field and let F(T) ∈ K[T] be a polynomial of degree greater than or equal to 1. Let HK(x) denote the height of x in K. Then there exists a positive constant CF, depending on F(T) only, such that for all x ∈ K we have that HK(x) ≤ CF·(HK(F(x)).

Proof. Since the case where the degree of F(T) is equal to 1 is obvious, we will assume that the degree of F(T) is greater than 1.