Topology of diophantine sets: remarks on Mazur's conjectures (Q2715538)

There are several conjectures on the topological closure of the set of \(\mathbb Q\)-rational points of a variety over \(\mathbb Q\), made by \textit{B. Mazur} [in: Galois representations in arithmetic algebraic geometry, Proc. Symp., Durham 1996, Lond. Math. Soc. Lect. Note Ser. 254, 239-265 (1998; Zbl 0943.14009)]. The authors prove that Mazur's conjecture for the real topology implies that there is no Diophantine model of the first order theory of \(\mathbb Z\) inside \(\mathbb Q\) in the language \(L_{\mathbb Z}\) of rings. For the case of global function fields, the authors show that the analogue of Mazur's conjecture does not hold, for which purpose they use a construction of \textit{T. Pheidas} [Invent. Math. 103, 1-8 (1991; Zbl 0696.12022)], but nevertheless, that for any prime power \(q\), the ring of polynomials \(\mathbb F_q [t]\) does admit a Diophantine model in the rational function field \(\mathbb F_q (t)\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00034].