Hilbert's Tenth Problem for function fields of varieties over C

TL;DR

This paper proves that Hilbert's Tenth Problem is undecidable for function fields of varieties over complex numbers, extending previous results from purely transcendental cases to more general varieties.

Contribution

It generalizes the undecidability of Hilbert's Tenth Problem from specific transcendental function fields to broader classes of function fields over complex varieties.

Findings

Hilbert's Tenth Problem is undecidable for function fields of varieties over C

Extends Kim and Roush's 1992 result to higher-dimensional varieties

Demonstrates undecidability in more general algebraic settings

Abstract

Let K be the function field of a variety of dimension at least 2 over an algebraically closed field of characteristic zero. 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 C(t_1,t_2) is undecidable.