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The Casas-Alvero conjecture for infinitely many degrees

Hans-Christian Graf von Bothmer·Oliver Labs·Josef Schicho·Christiaan van de Woestijne
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Abstract

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been some computational verifications for small degrees d. In this paper the conjecture is proved in the case where the degree of the polynomial is a power of a prime number, or twice such a power. Moreover, for each positive characteristic p, we give an example of a polynomial of degree d which is not a dth power but which has a common factor with each of its first d-1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.

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Hans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, Christiaan van de Woestijne. 2007-06-25. The Casas-Alvero conjecture for infinitely many degrees. https://doi.org/10.1016/j.jalgebra.2007.06.017

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