The Casas-Alvero conjecture for infinitely many degrees

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.Comment: 7 pages; v2: corrected some typos and references, and added section on computational aspect

Solving Diophantine problems on curves via descent on the jacobian

We suggest that the following plan will provide a powerful tool for trying to find the set of Q-rational points C(Q) on a curve C of genus>1:\ud (1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C.\ud (2) Deduce generators for J(Q) via an explicit theory of heights.\ud (3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).\ud We describe work just completed, which gives versions of (1),(2),(3) which are often workable in practice for genus 2, and outline the potential for a computationally viable generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have their own independent applications to other branches of the Mathematics of Computation, such as the search for large rank, the higher dimensional testing of well known conjectures, and algorithms for symbolic integration