# Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field

### Kirsten Eisenträger

The Pennsylvania State University, University Park, USA### Alexandra Shlapentokh

East Carolina University, Greenville, USA

## Abstract

We prove that the existential theory of any function field $K$ of characteristic $p>0$ is undecidable in the language of rings augmented by constant symbols for the elements of a suitable recursive subfield, provided that the constant field does not contain the algebraic closure of a finite field. This theorem is the natural generalization of a theorem of Kim and Roush from 1992. We also extend our previous undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of *any* function field of positive characteristic is undecidable in the language of rings without parameters.

## Cite this article

Kirsten Eisenträger, Alexandra Shlapentokh, Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 2103–2138

DOI 10.4171/JEMS/714