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**Existential definability with bounds on Archimedean valuations.**
*(English)*
Zbl 1068.03011

The area of the paper is “extensions of Hilbert’s tenth problem to number rings”, that is, the question whether, given a number field \(K\), there is an algorithm which determines the solvability of arbitrary diophantine equations in the ring of integers \({\mathcal O}_K\). It has been conjectured by Denef and Lipshitz that the answer to the question is negative, and all existing results to this day are negative, but only some cases are known (in particular, for abelian \(K\)). Since it became clear that some main tools (“Pell equations” and norm forms) of the first results are not applicable to the general case, researchers have been trying to find new ways of attacking the problem. Such a way has been suggested by Poonen and Cornelissen-Pheidas-Zahidi. The results of the paper present a new approach that may be critical for the future of this problem. A rough statement of the main results is: given a number field \(K\), if for some element \(\delta\), rational and quadratic over \(K\) (with some additional mild assumption), the graph over \({\mathcal O}_K\) of the relation “\(| \sigma (x)-\delta \sigma (y)| \leq | \sigma (z)| \) for all embeddings of \(K\) into C” is diophantine (a set is diophantine if it is definable by a formula which consists of existential quantifiers, followed by a diophantine equation) then the subset Z (rational integers) of \(K\) is diophantine over \({\mathcal O}_K\) and the analogue of Hilbert’s tenth problem for \({\mathcal O}_K\) has a negative answer.

Reviewer: Thanases Pheidas (Iraklion)

### MSC:

03B25 | Decidability of theories and sets of sentences |

11U05 | Decidability (number-theoretic aspects) |

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\textit{A. Shlapentokh}, J. Symb. Log. 68, No. 3, 860--878 (2003; Zbl 1068.03011)

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