Séminaire du 25 octobre 2010, 10h30: André Galligo, Laboratoire J. A. Dieudonné,
Université de Nice Sophia-Antipolis.
*Patterns in Roots of the Derivatives of a Random Polynomial*

I have associated to a real polynomial *f* the collection of all the
real roots of its derivatives organized in a 2D diagram (BD).
With a simple maple code, I observed that for many families of random
polynomial this set exhibited patterns with surprising structures.
Relying on fractional derivatives to interpolate this set, we get a
collection of continuous curves which on the picture *appear*
more regular. I called it the *stem* of the polynomial *f*.
These curves look like either line segments or ovals starting and
finishing at *x=0*;
each of them is related to one virtual roots of the polynomial *f*.
Patterns appear also for complex roots in the complex plane, in this
random setting
they are richer than Sendov conjecture on location of critical points.

Virginie Collette
Last modified: Mon Sep 20 14:24:06 CEST 2010