The CasasAlvero conjecture for infinitely many degrees
Abstract
Over a field of characteristic zero, it is clear that a polynomial of the form (Xa)^d has a nontrivial common factor with each of its d1 first derivatives. The converse has been conjectured by CasasAlvero. 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 d1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2006
 arXiv:
 arXiv:math/0605090
 Bibcode:
 2006math......5090G
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 12E05;
 13B25
 EPrint:
 7 pages