Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4587903 | Journal of Algebra | 2007 | 7 Pages |
Abstract
Over a field of characteristic zero, it is clear that a polynomial of the form d(X−α) 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 monic 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory