Hilbert's Tenth Problem over number fields, a survey
type: Item
instance of: scholarly article
zbMATH Open document ID: 0994.03001
author: Alexandra Shlapentokh
publication date: 2002-10-02T00:00:00Z
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 03D35
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 1607961
zbMATH Keywords: decidability
zbMATH Keywords: Hilbert's tenth problem
zbMATH Keywords: number field
zbMATH Keywords: diophantine equation
MaRDI profile type: Publication
title: Hilbert's Tenth Problem over number fields, a survey
review text: Hilbert's tenth problem asks for an algorithm which, given an arbitrary diophantine equation, checks the equation for having or not having integer solutions. The final answer to the problem is negative: no such algorithm exists. But what if one asks the similar question for the ring of integers of a number field, such as \(\mathbb{Z}[i]\) (the Gaussian integers)? Should an algorithm exist for any one such ring (therefore, by known arguments, also for any such ring that contains this one), Hilbert's tenth problem may be thought of as having an essentially positive answer -- with \(\mathbb{Z}\) being an exception. NEWLINENEWLINENEWLINEWork on the generalized problem has been initiated by Denef and Lipshitz, who produced a number of negative results and conjectured that the answer is in general negative. The author, who has herself produced similar results, presents an account of the existing results and organizes some of the known methods of proof in a way that a new researcher on this subject will appreciate.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00034].
reviewed by: Thanases Pheidas
description: scientific article; zbMATH DE number 1607961