Hilbert's Tenth Problem for Rings of Rational Functions

Notre Dame Journal of Formal Logic 43 (3):181-192 (2002)
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
Keywords diophantine problems   function fields   undecidability
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