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 $C=1$.

$\begingroup$ Take $XY = 0$. Conclude. $\endgroup$ – Hermès Mar 7 '17 at 20:13 — Hermès, Mar 7 '17 at 20:13
$\begingroup$ Oops! I want $C$ to be positive. $\endgroup$ – Jo Lasker Mar 7 '17 at 20:14 — Jo Lasker, Mar 7 '17 at 20:14

E. Joseph · Accepted Answer · 2017-03-07 20:16:56Z

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

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

answered Mar 7 '17 at 20:13

E. Joseph

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$\begingroup$ Sorry, I meant $C$ is positive. $\endgroup$ – Jo Lasker Mar 7 '17 at 20:15
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$\begingroup$ Thanks. I will accept when the site lets me in a few minutes. I wonder what nice conditions guarantee that the number of solutions is finite. $\endgroup$ – Jo Lasker Mar 7 '17 at 20:18
$\begingroup$ @JoLasker Your question made me wonder what happen if you impose $X_1,\ldots,X_n\in\mathbb N$. $\endgroup$ – E. Joseph Mar 7 '17 at 20:34

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