I have a nonlinear diophantine equation of the form $$g(x,y)t^4+h(x,y)t^3+w(x,y)t^2+f(x,y)t+d(x,y)=0$$ such that $t$ is a positive integer variable and $g(x,y),h(x,y),w(x,y),f(x,y),d(x,y)$ are polynomial functions in the positive integers variables $x,y$ and $$g(x,y)≠0,d(x,y)≠0$$ for all positive integers $x,y$
Then I am asking if one can consider this equation as a quartic one in $t$ and deduce that it has a finite number of solutions with respect to $t$ by using the fact that any polynomial function has a finite number of roots.
Of course for each $x$ and $y$ there are at most $4$ solutions, but allowing $x$ and $y$ to vary there's no reason to have only finitely many. For a trivial example, $t^4 - x^4$ has infinitely many solutions $t=x$.