Hilbert's tenth problem for function fields of varieties over algebraically closed fields of positive characteristic (Q1758239)

Let \(K\) be a function field of a variety of dimension \(\geq 2\) over an algebraically closed field \(k\) of characteristic \(p > 2\). It is proved that there exist elements \(z_1, z_2 \in K\) which are algebraically independent over \(k\) such that Hilbert's Tenth Problem for \(K\) with coefficients in \({\mathbb F}_p[z_1, z_2]\) is undecidable. The proof works by interpreting the ring of endomorphisms of an elliptic curve with division and two special pair relations in the given structure, by existential formulas. Endomorphisms are related to points on twists, and good twists are chosen using a theorem by Moret-Bailly. The result is both a generalisation and a week form of a result by Kim and Roush; week form because by Kim and Roush the two algebaric independent elements were canonic.