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

The univariate polynomial (X-α) 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=2p 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-α) 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 dth power and has nontrivial common factors with its d-1 first Hasse derivatives, which is a stronger condition than regular derivatives.

12E05Polynomials over general fields
12Y05Computational aspects of field theory and polynomials
12E20Finite fields (field-theoretic aspects)