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
they are richer than Sendov conjecture on location of critical points.
Last modified: Mon Sep 20 14:24:06 CEST 2010