Consider a special case of the Hilbert's 10th problem:

$f(\vec{x})=g(\vec{y})$, where $\vec{x}$ and $\vec{y}$ are **disjoint** ( i.e, the LHS and RHS do not have any common variables), moreover, $f$ and $g$ are polynomials with **positive** coefficients.

The question is, with the restrictions, whether the undecidability still holds?

For example, the Pell equation $x^2=3y^2+1$ falls into this class and is difficult to solve. I am wondering whether we can obtain undecidability outright.

[**Important note**: the Hilbert 10th question has two versions, ie., whether there is a solution in natural numbers, or in integers. In general, they are equivalent, but in this particular case, they are not. The answer below shows that checking whether there is a solution in integers is undecidable, but it is still open for the case that the solution must be natural numbers.]