The classical Thue equation is $$\displaystyle F(x,y) = h,$$ for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ is a non-zero integer. Typically, since $F$ is homogeneous, it is convenient to allow $h$ to be positive and further require that the inputs $(x,y)$ be relatively prime integers.

The fact that binary forms always split into linear forms over $\overline{\mathbb{Q}}$ is the reason why substantially better results can be proved for binary forms rather than general binary polynomials, and homogeneous polynomials in a greater number of variables.

Nevertheless, has any progress been made for equations of the form $$\displaystyle F(x_1, \cdots, x_n) = h,$$ where $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a homogeneous polynomial in $n$ variables?

I know that for $n = 3$ a 2009 paper by D.R. Heath-Brown has made progress on counting solutions inside the box $|x_i| \leq B$, with $h$ relatively small compared to $B$. O. Marmon generalized Heath-Brown's argument to the $n = 4$ case. Both employ a version of the determinant method pioneered by Bombieri and Pila in their 1989 paper. Both results have the deficit that the range of $h$ is fairly restricted. In particular, they are not able to prove the analogue of the following result for binary forms: if we wish to count the number of pairs $(x,y)$ such that $1 \leq F(x,y) \leq X$ with the assumption that $F(1,0) > 0$, then the exact order of magnitude is $X^{2/d}$ where $d = \deg(F)$.

Are there any other attempts to deal with the number of solutions to the higher dimensional analogue of Thue's equation? Any references would be appreciated.