Number of integer solution to $P(X_1,\dotsc,X_n)=c$ where $P$ is a homogeneous polynomial

Question

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 $C=1$.

Answer 1

This is not true without any conditions.

Take for instance $n=2$, and

$$P(X_1)={X_1}+X_2$$

which is homogeneous.

Then the equation $P(X_1,X_2)=0$ has infinitely many integer solutions:

$$(n,-n),\quad n\in \mathbb Z.$$

Edit.

Since you want $C>0$, you can take the equation $P(X_1,X_2)=1$, and the solutions are

$$(n,-n+1),\quad n\in \mathbb Z.$$