Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with integer coefficients has an integer solution.
However, if one restricts to various subsets of polynomials, then Hilbert's 10th admits a positive solution: for example, if one only looks at polynomials in one variable then such an algorithm does indeed exist: given $f(x) = a_d x^d + \cdots x_0, x_i \in \mathbb{Z}$ for $i = 0, \cdots, d$, this equation admits an integer solution if and only if $f(k) = 0$ for some divisor $k$ of $a_0$, so the algorithm reduces to factoring $a_0$ and then checking $f(k)$ for each $k | a_0$.
I believe Baker's theorem on linear forms in logs imply that an algorithm also exists for polynomials in two variables, but such an algorithm is highly impractical, due to the astronomical bounds Baker's method produces for the heights of possible solutions.
Hilbert's original question really only concerns hypersurfaces in $\mathbb{A}^n$ for $n \geq 1$. The previous two paragraphs show that for $n = 1,2$ the conclusion of Hilbert is actually true, while the first paragraph shows that the conclusion is false in general. My question is, for each $n \in \mathbb{N}$ what is the largest class of hypersurfaces in $\mathbb{A}^n(\mathbb{Z})$ for which the conclusion of Hilbert's 10th problem is true? Indeed, do we expect the conclusion of the MRDP theorem to hold as soon as $n \geq 3$?
