Let $f=f(x), g=g(x) \in \mathbb{C}[y][x] \subset \mathbb{C}(y)[x]$, where $\{x,y\}$ are variables over $\mathbb{C}$, and each of $\{f,g\}$ is of $x$-degree $\geq 1$, namely, $f$ and $g$ are non-constant polynomials over $\mathbb{C}(y)$ in one variable $x$.
Write $f=a_nx^n+\cdots+a_1x+a_0$, $g=b_mx^m+\cdots+b_1x+b_0$, where $a_i=a_i(y),b_j=b_j(y) \in \mathbb{C}[y]$, $1 \leq i \leq n, 1 \leq j \leq m$.
Fix $c \in \mathbb{C}$, and denote $\tilde{a_i}:=a_i(c)$ and $\tilde{b_j}:=b_j(c)$,
$\tilde{f}:=\tilde{a_n}x^n+\cdots+\tilde{a_1}x+\tilde{a_0}$ and
$\tilde{g}=\tilde{b_m}x^m+\cdots+\tilde{b_1}x+\tilde{b_0}$,
$1 \leq i \leq n, 1 \leq j \leq m$.
(Clearly, $\tilde{a_i},\tilde{b_j} \in \mathbb{C}$ so $\tilde{f},\tilde{g} \in \mathbb{C}[x]$).
Assume that $f$ and $g$ do not have a common zero in $\mathbb{C}(y)$ (= their resultant is non-zero).
Is it true that $\tilde{f}$ and $\tilde{g}$ do not have a common zero in $\mathbb{C}$?
Remarks: (1) Of course, if $f$ and $g$ do not have a common zero in $\mathbb{C}(y)$, then (in particular) $f$ and $g$ do not have a common zero in $\mathbb{C}$, but here I am asking about $\tilde{f}$ and $\tilde{g}$, not $f$ and $g$.
(2) Perhaps my question in trivial and has a positive answer? I only know that if $\tilde{f}$ and $\tilde{g}$ have a common zero $d \in \mathbb{C}$, then $f=f(x,y)|(x=d,y=c)=\tilde{f}(d)=0$ and $g=g(x,y)|(x=d,y=c)=\tilde{g}(d)=0$, but I do not see if this should imply that $f$ and $g$ have a common zero in $\mathbb{C}(y)$. (Perhaps a universal property is relevant here).
Any comments and hints are welcome!