Notre Dame Journal of Formal Logic

Hilbert's Tenth Problem for Rings of Rational Functions

Karim Zahidi

Abstract

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

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

Dates
First available in Project Euclid: 16 January 2004

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1074290716

Digital Object Identifier
doi:10.1305/ndjfl/1074290716

Mathematical Reviews number (MathSciNet)
MR2034745

Zentralblatt MATH identifier
1062.03019

Subjects
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]

Keywords
diophantine problems function fields undecidability

Citation

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. https://projecteuclid.org/euclid.ndjfl/1074290716


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