Let $P(X_1,\dotsc,X_n)$ be a homogeneous polynomial with integer coefficients. Let $C$ be an integer.
Is it true that the equation $P(X_1,\ldots,X_n)=C$ has only finitely many integer solutions?
EDIT: I meant $C$ is positive. Say $C=1$.
This is not true without any conditions.
Take for instance $n=2$, and
which is homogeneous.
Then the equation $P(X_1,X_2)=0$ has infinitely many integer solutions:
$$(n,-n),\quad n\in \mathbb Z.$$
Since you want $C>0$, you can take the equation $P(X_1,X_2)=1$, and the solutions are
$$(n,-n+1),\quad n\in \mathbb Z.$$