Subsets of all Diophantine's sets I have asked this question on math.stackexchange already:
https://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets
Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable $\Leftrightarrow$ graph of function is Diophantine.
Consider some subset $S$ of computable functions (for example some Grzegorczyk's class or polimomials-computable functions). There is subset of set of all Diophantine's set that is corresponded to $S$. 
Is it possible to describe  this subset easy in some cases?
 A: The elementary recursive functions are the functions with Diophantine graphs and elementary bounds on the new variables.  More precisely:
Let $f: N^k \rightarrow N$ be an elementary recursive function (Grzegorczyk class $\mathcal{E}^3$).
Then the graph $f(x_1,\ldots,x_k)=y$ is of the form
$$\exists z_1\cdots \exists z_j\thinspace p(x_1,\ldots,x_k,y,z_1,\ldots,z_j)=0 \wedge (\Sigma z_i) < 2^\wedge(\cdots 2^\wedge(\Sigma x_i) \cdots )$$
where $p$ is polynomial and the tower of 2s is of fixed length.
Furthermore, the projection of any graph of that form is an elementary recursive function.
The proof is just the standard proof for Hilbert's tenth problem, but keeping track of the bounds.
If you're happy with exponential Diophantine representations, this may be good enough.  (It also means you only need the Putnam-Davis-Robinson results to prove it.)  If you want to reduce this further to Diophantine representations, you can translate the exponential bounds into Diophantine bounds via Matiyasevich's results.
