*(English)*Zbl 1127.12002

The univariate polynomial ${(X-\alpha )}^{d}$ over a field of characteristic zero has a nontrivial factor with its $d-1$ first derivatives. The converse of this result has been conjectured by Casas-Alvero.

The authors show that the conjecture is true for some cases of $d$. More precisely, they show that, over a field of characteristic zero, for $d={p}^{k}$ or $d=2{p}^{k}$, $p$ a prime number, the only degree $d$ univariate monic polynomial that has nontrivial factors with its first $d-1$ first derivatives is $P={(X-\alpha )}^{d}$.

For fields of positive characteristic $p$, the authors present the counter-example $P={X}^{p+1}-{X}^{p}$ of degree $d=p+1$ which is not a $d$th power and has nontrivial common factors with its $d-1$ first Hasse derivatives, which is a stronger condition than regular derivatives.