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The Casas-Alvero conjecture for infinitely many degrees. (English) Zbl 1127.12002

The univariate polynomial ${\left(X-\alpha \right)}^{d}$ 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 $P={\left(X-\alpha \right)}^{d}$.

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.

##### MSC:
 12E05 Polynomials over general fields 12Y05 Computational aspects of field theory and polynomials 12E20 Finite fields (field-theoretic aspects)
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