Diophantine definability of Z over Q

Determine whether the ring of integers Z is Diophantine over the field of rationals Q; that is, establish whether there exists a polynomial F(t,x1,…,xn) with integer coefficients such that for every t in Q, t is in Z if and only if there exist x1,…,xn in Q satisfying F(t,x1,…,xn)=0.

Background

A classical route to showing the undecidability of Hilbert’s Tenth Problem over Q is to give a Diophantine definition of Z inside Q, which would allow encoding integer-existence questions within rational-existence questions. However, several deep conjectures in arithmetic geometry (e.g., Mazur-type conjectures) are believed to preclude such definability, leaving the status of this definability problem as a major unresolved question.

The paper explicitly mentions the existence of conjectures implying non-existence of such a definition and highlights the (non)existence question as a major open issue in the area.