$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns

Let Z be the ring of integers. Hilbert’s Tenth Problem (HTP), the tenth one of his 23 famous mathematical problems presented in the 1900 ICM, asks for an algorithm to determine for any given polynomial P (x1, . . . , xn) ∈ Z[x1, . . . , xn] whether the diophantine equation P (x1, . . . , xn) = 0 has solutions x1, . . . , xn ∈ Z. This was solved negatively by Yu. Matiyasevich [10] in 1970, on the basis of the important work of M. Davis, H. Putnam and J. Robinson [6]; see also Davis [5] for a nice introduction. Z.-W. Sun [15] proved his 11 unknowns theorem which states that there is no algorithm to determine for any P (x1, . . . , x11) ∈ Z[x1, . . . , x11] whether the equation P (x1, . . . , x11) = 0 has solutions over Z. Let Q be the field of rational numbers. It remains open whether HTP over Q is undecidable. However, J. Roboinson [14] used the theory of quadratic forms to prove that one can characterize Z by using the language of Q in the following way: For any t ∈ Q we have t ∈ Z ⇐⇒ ∀x1∀x2∃y1 · · · ∃y7∀z1 · · · ∀z6[f(t, x1, x2, y1, . . . , y7, z1, . . . , z6) = 0],