I consider for a constant $k\in\mathbb{N}$ the following Diophantine equation
$$\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - k = 0$$
over the hypercube $C^n=[0,m]^n$ for some $m\in\mathbb{N}$. I am trying to understand whether there are integer solutions to this equation inside $C^n$. The basic approach would be to check whether there are any solutions $\mod p$ for any prime $p$ and in case that there is a $p$ such that there are not, to refute the existence of integer solutions. But this does not exploit the restriction that I am only looking in $C^n$.
If $p>\max_{(x_1,...,x_n)\in C^n} (\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - m )$, then the restricted equation $\mod p$ has the same number of solutions as without $\mod p$. So, in any case, I only have to consider the finite number of $p$ which are smaller than the $\max$ mentioned before.
Basically, assuming $p_0$ is the smallest prime such that $\max_{(x_1,...,x_n)\in C^n} (\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - m )<p_0$, the "restricted" $L$ function associated to these point counts $(a_n)$ equals
$$L(X,s)= g(s) + a_{p_0}(\zeta(s)-q(s))$$
where $g$ is some rational function, defined by the point counts for $p<p_0$ and
$$q(s) :=\sum_{n=1}^{p_0-1}\frac{1}{n^{s}}.$$
Can I use this in any way to say something about the number of solution of the Diophantine equation of interest here? Or is there any theory on restricted Diophantine equations, which I could exploit?