The Casas-Alvero conjecture for infinitely many degrees
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.
Cite
@article{arxiv.math/0605090,
title = {The Casas-Alvero conjecture for infinitely many degrees},
author = {Hans-Christian Graf von Bothmer and Oliver Labs and Josef Schicho and Christiaan van de Woestijne},
journal= {arXiv preprint arXiv:math/0605090},
year = {2009}
}
Comments
7 pages; v2: corrected some typos and references, and added section on computational aspects