Let $K$ be a field, $F = K(T)$ a rational function field over $K$. Let $G \in F[C, X_1, \ldots, X_n]$ be a polynomial with coefficients in $F$.
We can consider the polynomial $G(c, X_1, \ldots, X_n)$ for several values of $c \in K$. For some values of $c \in K$, $G(c, X_1, \ldots, X_n)$ might have a zero in $F$, whereas for others it might not. Can we find a natural number $m \in \mathbb{N}$ depending on $G$ and $K$ such that, for all $c \in K$ for which $G(c, X_1, \ldots, X_n)$ has a zero in $F$, it has a zero $(x_1/q, \ldots, x_n/q)$ in $F$ where $x_1, \ldots, x_n, q \in K[T]$ have degree bounded by $m$?
An explicit construction which finds such an $m$ would be awesome, but I would also be perfectly happy with some compactness argument which only shows the existence.