We show that if R is a nonconstant regular (semi-)local
subring of a rational function field over an algebraically closed field of
characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer;
that is, there is no algorithm to decide whether an arbitrary Diophantine equation
over R has solutions over R or not. This result can be seen as evidence
for the fact that the corresponding problem for the full rational field is also unsolvable.
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