Hilbert’s Tenth Problem for function fields of varieties over algebraically closed fields of positive characteristic

Let K be the function field of a variety of dimension ≥ 2 over an algebraically closed field of odd characteristic. Then Hilbert’s Tenth Problem for K is undecidable. This generalizes the result by Kim and Roush from 1992 that Hilbert’s Tenth Problem for the purely transcendental function field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\overline{\mathbb{F}}_p}}(t_1,t_2)}$$\end{document} is undecidable when p > 2.