Undecidability in function fields of positive characteristic

TL;DR

This paper proves that the first-order theory of function fields of positive characteristic is undecidable in the language of rings, extending to characteristic 2 for certain cases, using elliptic curve rank results.

Contribution

It establishes undecidability of the first-order theory for a broad class of function fields, including characteristic 2, using novel applications of elliptic curve rank results.

Findings

Undecidability of the first-order theory in characteristic p>2

Undecidability extends to characteristic 2 for certain function fields

Uses elliptic curve ranks to prove logical undecidability

Abstract

We prove that the first-order theory of any function field K of characteristic p>2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.