We consider the modular GCD-algorithm for $F[x, y]$.
INPUT: $f, g \in F[x, y]$ primitive as polynomials in $F[y][x]$, with $\deg_x f \geq \deg_x g$; $d \in \mathbb{N}$ with $d \geq \deg_y(f), \deg_y(g)$. OUTPUT: $h = \gcd(f, g)$.
Let $b = \gcd(\operatorname{lc_x}(f), \operatorname{lc_x}(g))$. Choose a monic irreducible $g \in F[y]$ with $\deg p = d + 1 + \deg b$. Let $\overline{\phi}$ denote the result of reducing mod $p$ a polynomial $\phi \in F[x, y]$, i.e., $$F[y][x] \rightarrow F[y]/\langle p \rangle[x], \quad \phi = \sum_j \phi_j(y)x^j \mapsto \sum_j \phi_j(y) + \langle p \rangle x^j = \overline{\phi}$$
Compute $w \in F[x, y]$, $\deg_y w < \deg p$ with $\overline{w} = \overline{b} \cdot \gcd(\overline{f}, \overline{g})$.
Since $\overline{w} | \overline{f b}$, $\overline{w} | \overline{g b}$, we may compute $f^\star, g^\star \in F[x, y]$ with $\deg_y(f^\star), \deg_y(g^\star) < \deg p$ and $\overline{f^\star} = \overline{f b}/\overline{w}$ and $\overline{g^\star} = \overline{g b}/\overline{w}$.
Then the halting condition is $\deg_y(f^\star w) = \deg_y(f b)$ and $\deg_y(g^\star w) = \deg_y(g b)$.
Prove that the halting condition holds if and only if $p$ does not divide the $x$-resultant of $f/h, g/h$, i.e., (halting condition) $\Leftrightarrow \neg p \mid \text{res}_x(f/h, g/h)$
-- So far I have only tried for the implication to suppose that $p$ does divide the resultant, so $\text{res}_x(f/h, g/h) \equiv 0 (\mod{p})$ and so there exist polynomials $s,t \in F[x]$ with $s\frac{f}{h}+t\frac{g}{h} \equiv 0 (\mod{p})$ with $\deg_x s < \deg_x g/h,\deg_x t < \deg_x f/h$ but I can't seem to find the contradiction.
Any help is appreciated!