## 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

#### 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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