8 - Diophantine undecidability of function fields

Summary

Fields of positive characteristic do not contain integers, and therefore constructing Diophantine definitions of integers to establish Diophantine undecidability, as we have done for some number rings, is not an option here. However, function fields over finite fields of constants do possess Diophantine models of integers, a fact which will allow us to show that the analog of Hilbert's Tenth Problem is undecidable over these fields. It will take us some time to arrive at the desired results and we will start with a seemingly unrelated point.

Before proceeding with our investigation we should note that the main ideas presented in this chapter are due to Cornelissen, Eisenträger, Pheidas, Videla, Zahidi, and the present author, and can be found in [6], [22], [67], [66], [69], [98], [102], and [117].

Defining multiplication through localized divisibility

This section contains some technical definability results which will allow us to make a transition from characteristic 0 to positive characteristic. The original idea underlying this method belongs to Denef (see [17]) and Lipshitz (see [48]–[50]). It was developed further by Pheidas in [66]. We reproduce Pheidas's results below.

We start with fixing notation and a definition.

Notation 8.1.1. In this section p will denote a fixed rational prime.