The relative undecidability conjecture for finiteness of integral points

An integral Diophantine equation is an equation

$$f(x_1,x_2,\ldots,x_n)=0$$

where $f(x_1,\ldots,x_n)$ is a polynomial with integral coefficients. The existence problem asks whether the equation has an integral solution, while the finiteness problem asks whether its set of integral solutions is finite.

Relative undecidability conjecture. The finiteness problem for integral points is undecidable relative to the existence problem.

This conjecture concerns the computational relationship between deciding whether integral points exist and deciding whether there are only finitely many of them. The source presents it as a framework for studying relative computability questions beyond the undecidability of existence alone.