## Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent (1999)

Venue: | of Contemp. Math |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Cornelissen99topologyof,

author = {Gunther Cornelissen and Karim Zahidi},

title = {Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent},

booktitle = {of Contemp. Math},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.

### Citations

538 | Model theory
- Hodges
- 1993
(Show Context)
Citation Context ..., ci) where ci are primary predicates (“constants”), less in number than |R|. We call LR a ring language. We define LZ = (+, ·, 0, 1) and Lt = (+, ·, 0, 1, t) for any t ∈ R. Example 1.2. (Tarski, cf. =-=[9]-=-, pp. 202–206) (a) An algebraically closed field k admits elimination of quantifiers in the language LZ. Hence any LZ-definable subset in k n is a boolean combination of sets defined by an equation. T... |

238 |
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern
- Faltings
- 1983
(Show Context)
Citation Context ...(a) Conjecture 2.1 is true for curves V . One can assume V to be projective and non-singular. The case where V has genus g ≥ 2 is settled by Faltings’s theorem, which says that V (Q) is a finite set (=-=[6]-=-). If V has genus 0, then either V (Q) is empty, or V is Q-birational to A 1 , and A 1 (Q) is topologically dense in A 1 (R). Finally, assume that V has genus 1. It is known that V (R) is isomorphic t... |

55 |
Hilbert's Tenth Problem. Diophantine Equations: Positive Aspects of a Negative Solution, in: Mathematical developments arising from Hilbert
- Davis, Matiyasevich, et al.
- 1976
(Show Context)
Citation Context ...(b)), one can find a diophantine model of the integers (Z, LZ) in (R, LR), and then rely on the fact that the diophantine theoryREMARKS ON MAZUR’S CONJECTURES 5 of the integers is undecidable ([13], =-=[3]-=-). It has been suggested that, with this more flexible definition, one would be able to find a diophantine model of the integers in the rationals: Question 3.5. Does (Z, LZ) admit a diophantine model ... |

49 |
On definable subsets of p-adic fields
- Macintyre
- 1976
(Show Context)
Citation Context .... This gives a nice description of the definable subsets of R: they are finite unions of intervals. (c) More examples in the same vein exist, e.g., a description of definable sets over p-adic fields (=-=[12]-=-, [5]), or generalization of (R, L≥) via o-minimal expansions. (d) To give an example with a different language, existentially definable sets of Z in the language (0, 1, +, |) are unions of arithmetic... |

37 |
Effective procedures in field theory
- Fröhlich, Shepherdson
- 1956
(Show Context)
Citation Context ...al functions. More precisely, Fq[t] is a recursive ring (cf. Rabin [20]), because Fq is recursive (since finite), and hence the same holds for the polynomial ring over Fq (cf. Fröhlich and Sheperdson =-=[7]-=-). So there exists an injective map θ : Fq[t] → Z≥0 such that the graphs of addition and multiplication are recursive on Z≥0, and hence (θ(Fq[t]), θ(Lt)) is a diophantine model of (Fq[t], Lt) in (Z≥0,... |

33 |
p-adic semi-algebraic sets and cell decomposition
- Denef
- 1986
(Show Context)
Citation Context ... gives a nice description of the definable subsets of R: they are finite unions of intervals. (c) More examples in the same vein exist, e.g., a description of definable sets over p-adic fields ([12], =-=[5]-=-), or generalization of (R, L≥) via o-minimal expansions. (d) To give an example with a different language, existentially definable sets of Z in the language (0, 1, +, |) are unions of arithmetic prog... |

32 | The topology of rational points
- Mazur
- 1992
(Show Context)
Citation Context ...ragraph, we comment upon a non-archimedean version of this conjecture. Though most of these observations are folklore, they do not seen to have been written down previously. 2. Mazur’s conjectures In =-=[14]-=-, [15] and [16], Barry Mazur has proposed and discussed several conjectures and questions about the behaviour of the set of Q-rational points of a variety over Q under taking topological closure w.r.t... |

22 |
Double fibres and double covers: paucity of rational points
- Colliot-Thélène, Skorobogatov, et al.
- 1997
(Show Context)
Citation Context ...4]). Remark 2.3. Mazur has made even stronger conjectures, some of which had to be slightly modified, due to the construction of a counterexample by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer (=-=[1]-=-). For an extensive (unsurpassable) exposition and more examples, we refer to the original sources [14], [15] and [16]. We will concentrate on the model-theoretical aspects of the conjectures, which a... |

21 | The diophantine problem for addition and divisibility - Lipshitz - 1978 |

16 |
The Diophantineness of Enumerable Sets
- Matijasevic
(Show Context)
Citation Context ...n (3.2(b)), one can find a diophantine model of the integers (Z, LZ) in (R, LR), and then rely on the fact that the diophantine theoryREMARKS ON MAZUR’S CONJECTURES 5 of the integers is undecidable (=-=[13]-=-, [3]). It has been suggested that, with this more flexible definition, one would be able to find a diophantine model of the integers in the rationals: Question 3.5. Does (Z, LZ) admit a diophantine m... |

14 |
The Diophantine problem for polynomial rings and fields of rational functions
- Denef
- 1978
(Show Context)
Citation Context ...ing the relation “multiplication”. For an example, consider the proof that (Z, LZ) admits a diophantine model in (R := S[t], Lt) for any commutative unitary domain S of characteristic zero, see Denef =-=[4]-=-. He takes for G the torus G √ m,R[ ∆] of discriminant ∆ = t2 − 1, which is non-split over R; G(R) has rank one: any R-point is given by a solution (xn, yn) to the Pell-equation X2 − ∆Y 2 = 1 (i.e., a... |

14 |
Speculations about the topology of rational points: an update, Astérisque
- Mazur
- 1995
(Show Context)
Citation Context ...h, we comment upon a non-archimedean version of this conjecture. Though most of these observations are folklore, they do not seen to have been written down previously. 2. Mazur’s conjectures In [14], =-=[15]-=- and [16], Barry Mazur has proposed and discussed several conjectures and questions about the behaviour of the set of Q-rational points of a variety over Q under taking topological closure w.r.t. some... |

14 |
Open problems regarding rational points on curves and varieties
- Mazur
- 1998
(Show Context)
Citation Context ...ment upon a non-archimedean version of this conjecture. Though most of these observations are folklore, they do not seen to have been written down previously. 2. Mazur’s conjectures In [14], [15] and =-=[16]-=-, Barry Mazur has proposed and discussed several conjectures and questions about the behaviour of the set of Q-rational points of a variety over Q under taking topological closure w.r.t. some metric i... |

8 |
Some remarks on the Diophantine problem for addition and divisibility
- Lipshitz
- 1980
(Show Context)
Citation Context ... via o-minimal expansions. (d) To give an example with a different language, existentially definable sets of Z in the language (0, 1, +, |) are unions of arithmetic progressions (a result of Lipshitz =-=[11]-=-). The moral is that if the (existentially) definable sets for such M have a sufficiently easy description, then the first-order (respectively, existential) theory of M is decidable – this is the case... |

4 |
Stockage diophantien et hypothèse abc généralisée
- Cornelissen
- 1999
(Show Context)
Citation Context ...hantine. A similar notion of (positive) existential model exists. Example 3.2. (a) If (M2, L) admits a diophantine model in (M, L), then the latter structure is said to admit diophantine storing (cf. =-=[2]-=-). This is true, for example, for (Z, LZ). For non-algebraically closed rings (R, LR) admitting diophantine storing, one can always choose k = 1 in the above definition. For if (M ′, L ′, φ ′) admits ... |