Hilbert's Tenth Problem over number fields, a survey (Q2715530)

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].