Hilbert's tenth problem for rings of rational functions (Q1430905)

Let \(R\) be a non-constant regular semilocal subring of a field of rational functions \(F(t)\) over an algebraically closed field \(F\) of characteristic \(0\). It is proved that the analogue of Hilbert's tenth problem for \(R\) (does there exist an algorithm which determines the solvability in \(R\) of systems of multivariate polynomial equations with coefficients in \(\mathbb Z[t]\)?) has a negative answer.