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.
"Hilbert's Tenth Problem for Rings of Rational Functions." Notre Dame J. Formal Logic 43 (3) 181 - 192, 2002. https://doi.org/10.1305/ndjfl/1074290716