Notre Dame Journal of Formal Logic

Hilbert's Tenth Problem for Rings of Rational Functions

Karim Zahidi


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.

Article information

Notre Dame J. Formal Logic, Volume 43, Number 3 (2002), 181-192.

First available in Project Euclid: 16 January 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]
Secondary: 11U05: Decidability [See also 03B25] 12L05: Decidability [See also 03B25]

diophantine problems function fields undecidability


Zahidi, Karim. Hilbert's Tenth Problem for Rings of Rational Functions. Notre Dame J. Formal Logic 43 (2002), no. 3, 181--192. doi:10.1305/ndjfl/1074290716.

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