A polynomial in the complex variable $z$, whose coefficients are themselves complex polynomials in another complex variable $a$, looks like $$ f\in\Bbb C[A][Z],\;\;f(z,a)=c_0(a)+c_1(a)z+\cdots+c_n(a)z^n $$ with $c_j\in\Bbb C[A]$ of degree less or equal than $1$. Assume not all $c_j$ are constant and the leading coefficient $c_n\equiv1$.
I need to prove that the roots of $z\mapsto f(z,a)$ are simple for all $a\in\Bbb C$ except for a finite number of $a$.
A root of $f(\cdot,a)$ is simple if and only if it is not a root of $f'(\cdot,a)$ and this happens if and only if the resultant $P(a)$ of $f(\cdot,a),f'(\cdot,a)$ doesn't vanish.
Now $P\in\Bbb C[A]$, hence if it is not constantly $0$, then the claim is true (if $P$ is non constant then it vanishes for a finite number of values of $a$, while if it is constant and non zero, $f(\cdot,a)$ and $f'(\cdot,a)$ don't share a zero for any $a$).
Is it always possible to exclude the case $P\equiv0$?
