I am studying ruled surfaces in $\mathbb{R}^{3}$ and I am trying to compare the Gaussian curvature of two such surfaces constructed along the same curve. In order to do so, I end up considering an expression of the form $$ H(t,u)=\frac{f(t,u)}{g(t,u)}, $$ where both $f(t,u)$ and $g(t,u)$ are nonzero polynomials of equal degree in $u$.
I would like to know whether there is a simple way to examine the dependence of $H$ on $u$. In other words, under what conditions on numerator and denominator is the ratio independent of $u$?
Taking the derivative is not an option because the expression becomes really messy.
It's not clear to me what sort of answer is useful to you. If you have explicit real polynomials $f(t, u), g(t, u)$ whose ratio is independent of $u$, then you could factor them over $\mathbb{R}[t, u]$ (or a computable subfield...), and all the irreducible factors that actually involve $u$ must cancel.
Why? In this case we have $f(t,u)/g(t,u) = f(t,0)/g(t,0)$, so $f(t,u) g(t,0) = g(t,u) f(t,0)$. As stated this is an equality on the level of functions, but since $\mathbb{R}$ is infinite, it's also an equality on the level of formal polynomials. Now take an irreducible factor $p(t, u)$ of $f(t, u)$ that actually depends on $u$. By unique factorization, $p(t, u)$ divides either $g(t, u)$ or $f(t, 0)$. Since it can't divide $f(t, 0)$, it must divide $g(t, u)$.
Another way to say it is that there are polynomials $\alpha(t), \beta(t), h(t, u)$ such that $f(t, u) = \alpha(t) h(t, u)$ and $g(t, u) = \beta(t) h(t, u)$. Note that the obvious constraint that the $u$-degree of $f$ and $g$ is equal is clear here.