Consider the hyperplane $$X:=V(a_0x_0+a_1x_1+\cdots+a_Nx_N+a_{N+1}x_{N+1}) \subset \mathbb P^{N+1},$$ where $a_0,a_1, \cdots, a_{N+1} \in \mathbb Q$ are not all zero. Define the height function as follows: $$H(P)=\max \{|x_0|,|x_1|, \cdots, |x_N|\},~P=(x_0, \cdots, x_{N}) \in \mathbb P^N(\mathbb Q).$$
I need to prove that for each integer $M \geq 1$, $$\#\{P \in X(\mathbb Q): H(P) \leq M \} \leq C(2M+1)^{(N+1)}$$ for some constants $C>0$.
At first we recall the following lemma:
Lemma (Northcott property): Fix an integer $n \geq 1$, there are at most $(2M+1)^{(n+1)}$ of points in $\mathbb P^n(\mathbb Q)$ of height $M$.
To prove the statement, it is sufficient to show that $$X \cong \mathbb P^N(\mathbb Q).$$ Then by using the above lemma and taking $C=1$, the statement will follow.
So how to construct an isomporphism between the hyperplane $X$ and the projective space $\mathbb P^{N}(\mathbb Q)$, where $X$ lies in the ambient space $\mathbb P^{(N+1)}(\mathbb Q)$.
Let us recall that the notation $[x_0:x_1: \cdots:x_N]$ stands for equivalence class of points in $\mathbb P^N(\mathbb Q)$. Two points $(x_0, \cdots, x_N)$ and $(y_0, \cdots, y_N)$ are equivalent if $$(x_0, \cdots, x_N)=\lambda (y_0, \cdots, y_N)$$ for some non zero element $\lambda \in \mathbb Q$.
So if I define a map $f: X \to \mathbb P^N(\mathbb Q)$ by $$g(x_0, \cdots, x_N,x_{N+1})=(\frac{x_0}{x_{N+1}}: \frac{x_1}{x_{N+1}}, \cdots, \frac{x_N}{x_{N+1}}).$$ This seems injective.
But we need surjectivity as well. Does the above map help ?