On the Difference of 4-Gonal Linear Systems on some Curves

Let C = (C, g^1/4 ) be a tetragonal curve. We consider the scrollar invariants e1 , e2 , e3 of g^1/4 . We prove that if W^1/4 (C) is a non-singular variety, then every g^1/4 ∈ W^1/4 (C) has the same scrollar invariants.

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.