Suppose, we have a system of diophantine equations and also restrictions to the variables such as $0\le a\le x$ that can also be inequalities.
Can we transform this system in a diophantine equation that has a solution if and only if the given system has a solution ? In other words, is the system equivalent to some diophantine equation ?
An equation system can be considered by squaring terms and add them, but I am not sure how inequalities can be considered.
If you want to impose the condition that $x \ge y$ where $x$ and $y$ are integers (or rational numbers) you can introduce four more variables and then ask that
$$x-y = a^2 + b^2 + c^2 + d^2.$$.
Clearly this equality implies the LHS is positive. OTOH every non-negative integer (respectively rational) is the sum of four integral (resp. rational) squares, so there are always solutions for the new variables as long as the inequality is satisfied.
You can do strict inequalities as well: this is only required in the rational case (since for integers $x > y$ is the same as $x \ge y + 1$), and in the rational case one introduces a further variable and demands that
$$(x-y)e = 1.$$