Let's define the following languages over the alphabet $\Sigma=\{0,1\}$:
H10 is the language of all strings that are encoding of diophantine polynomial equation with integer coefficients and $n$ unknowns/variables that has at least 1 integers root/solution, i.e. there exists a root/solution in $\Bbb{Z}^n$ that solves the diophantine polynomial equation with integer coefficients with $n$ unknowns/variables.
RH10 is the language of all strings that are encoding of diophantine polynomial equation with integer coefficients and $n$ unknowns/variables that has at least 1 integers root/solution, each integer in the closed interval $[-2^n,2^n]$ or in other words there exists a root/solution in $[-2^n,2^n]^n$ that solves the diophantine polynomial equation with integer coefficients with $n$ unknowns/variables.
If RH10 can be solved in deterministic polynomial time, does this imply that H10 is decidable? If there is a polynomial-time algorithm for RH10, then it can't be enumerating all possible solutions in $[-2^n,2^n]^n$ (there are too many of them to try in polynomial time), so it seems like it must somehow find solutions regardless of how large they are, and thus also be able to solve H10 too. Is that right? Does it follow from this that $P$ is unequal to $NP$, as H10 is known to be, in fact, undecidable?