Given a polynomial $f(z)\in\mathbb{C}[z]$, there exists only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots?

Question

Given a polynomial $f(z)\in\mathbb{C}[z]$, $\exists$ only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots? Is above true in general? Is it true for polynomials of the form $f(z) = (z-z_1)\cdot ... \cdot (z - z_n)$ where $z_1, ... , z_n \in \mathbb{C}$are distinct?

Answer 1

A polynomial has repeated roots iff its discriminant is zero. The discriminant is a polynomial in its coefficients. So the discriminant of $f(z)-c$ is a polynomial, say $D(c)$, in $c$. Either $D(c)$ has finitely many zeros, which is what we want, or it is identically zero. In that case $f(z)=c$ has repeated root for all $c$. But that's not true. If $f(z)$ has leading term $z^n$, then when $|c|$ is large, the roots of $f(z)=c$ approximate those of $z^n=c$ which are distinct.

Answer 2

A multiple root of $f(z)-c$ is at the same time a root of $f'(z)$. As the polynomial $f'$ has only finitely many roots, the claim follows. The only special case is when $f'\equiv 0$, and indeed then $f$ is constant and for one specific value of $c$ has roots at all.