polynomials – A Generalization of the Casas-Alvero Conjecture

The Casas-Alvero conjecture claims that over a characteristic 0 field, any degree $d$ polynomial $f(x)$ such that, for all $0 leq i leq d-1$, $$gcd(f,f^{(i)}) neq 1$$ must have exactly one root.

I’m proposing a generalization of this conjecture: Suppose there exists $n leq d$ such that for all $1 leq j < n$, $$gcd(f, f^{(d-j)}) neq 1.$$ Then $f$ does not have exactly $n$ roots. This would imply Casas-Alvero by a simple induction argument.

I have proven this conjecture in $mathbb{C}$ for $n=2$ and $3$, but $n=4$ seems quite a bit harder. Perhaps there is a simple counterexample, but my programming skills are not up to the task to search for it. (I’d be happy to provide my proof in these two cases to those who are interested – it uses the Gauss-Lucas theorem and is in my opinion quite pleasing.)