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

Shlapentokh, Alexandra

J. Symb. Log. 68, No. 3, 860-878 (2003).

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)

Keywords:

decidability; Hilbert’s tenth problem; number ring

References:

[1]	Jean-Louis Colliot-Thélène, Alexei Skorobogatov, and Peter Swinnerton-Dyer Double fibres and double covers: Paucity of rational points , Acta Arithmetica , vol. 79 (1997), pp. 113–135. · Zbl 0863.14011
[2]	Martin Davis Hilbert’s tenth problem is unsolvable , American Mathematical Monthly , vol. 80 (1973), pp. 233–269. · Zbl 0277.02008
[3]	Martin Davis, Yurii Matijasevich, and Julia Robinson Positive aspects of a negative solution , Proceedings of Symposia in Pure Mathematics , vol. 28, American Mathematical Society,1976, pp. 323– 378. · Zbl 0346.02026
[4]	Jan Denef Hilbert’s tenth problem for quadratic rings , Proceedings of the American Mathematical Society , vol. 48 (1975), pp. 214–220. · Zbl 0324.02032
[5]	Jan Denef and Leonard Lipshitz Diophantine sets over some rings of algebraic integers , Journal of London Mathematical Society , vol. 18 (1978), no. 2, pp. 385–391. · Zbl 0399.10049
[6]	Gerald Janusz Algebraic number fields , Academic Press, New York,1973. · Zbl 0307.12001
[7]	Helmut Koch Number theory. algebraic numbers and functions , American Mathematical Society, Providence, Rhode Island,2000.
[8]	Barry Mazur The topology of rational points , Experimental Mathematics , vol. 1 (1992), no. 1, pp. 35–45. · Zbl 0784.14012
[9]	Thanases Pheidas Hilbert’s tenth problem for a class of rings of algebraic integers , Proceedings of American Mathematical Society , vol. 104 (1988), no. 2, pp. 611–620. · Zbl 0697.12020
[10]	Bjorn Poonen Using elliptic curves of rank one towards the undecidability of Hilbert’s Tenth Problem over rings of algebraic integers , Algorithmic number theory (C. Fieker and D. Kohel, editors), Lecture Notes in Computer Science, vol. 2369, Springer Verlag,2002, pp. 33–42. · Zbl 1057.11068
[11]	Julia Robinson Definability and decision problems in arithmetic , Journal of Symbolic Logic, vol. 14 (1949), pp. 98–114. JSTOR: · Zbl 0034.00801
[12]	Harold Shapiro and Alexandra Shlapentokh Diophantine relations between algebraic number fields , Communications on Pure and Applied Mathematics , vol. XLII (1989), pp. 1113–1122. · Zbl 0698.12022
[13]	Alexandra Shlapentokh Extension of Hilbert ’s tenth problem to some algebraic number fields, Communications on Pure and Applied Mathematics , vol. XLII (1989), pp. 939–962. · Zbl 0695.12020