Consider the diophantine equation three variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a linear polynomial in $y,z$ and $L_1(y),L_2(z)$ are linear in $y,z$ respectively.
In general such an equation is difficult to solve if $x,y,z$ are independent.
However suppose that there are univariate linear functions $L'(y)$ and $L''(z)$ such that if $(x^*,y^*,z^*)$ is a solution to the three variable diophantine equation then
$GCD(L'(y^*),L''(z^*)=1$
There exists integers $1<a,b$ with $a|L'(y^*)$ and $b|L''(z^*)$ such that $ab=x^*$
holds always. Then the unknown $x$ depends on $y,z$ and so in principle we should be able to eliminate $x$ and make this as a two variable quadratic diophantine equation.
One cannot say there is no small degree algebraic relation between $x$ and $y,z$ since we have already provided one by the quadratic equation. Unless we provide a different relation we cannot use elimination theory. However there is arithmetic relation. There is some hope. Is there a way to at least in principle reduce the problem to solving two unknowns through some functions coming from arithmetic?
What is the minimal degree of any other polynomial relation between $x,y,z$?
It is unclear if either of two is possible
I. There is an explicit arithmetic function that relates $x$ and $y,z$.
II. There is any other low degree polynomial relation between $x$ and $y,z$. Note that by formulating the problem itself we have provided a quadratic relation between $x$ and $y,z$. So there is a polynomial relation. The problem is if there is another polynomial relation.