= Rational multiplicity bound for a linear auxiliary polynomial
Let $F(X,Y)=P(X)+YQ(X)$ with $P,Q\in\mathbb Z[X]$ linearly independent, $\deg_XF\leq n$, and $H(F)\leq C^n$. For every $\delta>0$, if a reduced rational $p/q$ has sufficiently large denominator in terms of $C$ and $\delta$, then $F(X,y)$ has multiplicity at most $\delta n+1$ at $p/q$, for every fixed real $y$.
Indeed, the nonzero <Wronskian>
$$
W=PQ'-P'Q
$$
has degree below $2n$ and height at most $C_0^n$. A zero of multiplicity $r$ of $F(X,y)$ forces a zero of multiplicity at least $r-1$ of $W$. By <Gauss lemma for polynomials>, $(qX-p)^{r-1}$ then divides $W$ in $\mathbb Z[X]$, so $q^{r-1}$ divides the leading coefficient of $W$ and is at most $C_0^n$.
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