Let $f∈Q[X]$ and not constant or of the form $(x−a)^n$. Suppose:


$f\_1:=\frac{f}{gcd(f,D^2f)}$


and


$f\_2:=\frac{f\_1}{gcd(f\_1,Df\_1)}$


where $Df$ stands for the formal derivative.


Is it true that $gcd(f\_2,Df\_2)=gcd(f\_2,D^2f\_2)=1$ ?



Generating polynomials that are co-prime to their first and second derivatives
No.


Let
$$f(x)=(x-3x^3)(x+1/3).$$
Then $f''(-1/3)=0$ and
$$f\_1(x)=x-3x^3.$$
Since $f\_1$ and $f\_1'$ have no common roots, $f\_2=f\_1$. But $gcd(f\_2,D^2f\_2)=x$.


