Roots of polynomials with integer coefficients

[John Baez]

On my homepage you can see a picture drawn by Dan Christensen,
showing roots of all polynomials of degree less than or equal to 5
with integer coefficients ranging from -4 to 4:

http://math.ucr.edu/home/baez/

There are lots of interesting patterns. Has anyone studied this
sort of picture?

[ogerard]

Hello John,

You can look at Bayley et al work on polynomial roots
(they were focused on -1,1 coefficients and more on global properties
such as density, orientation, approximation paths than on local
patterns)

http://crd.lbl.gov/~dhbailey/expmath/

You will find a page devoted to that in his frequent co-author's home

http://oldweb.cecm.sfu.ca/personal/loki/Projects/Roots/Book/

and it was included in the numerous topics of
the series of two books on Experimental Mathematics.
(I suggest a look at the journal published by AKPeters
with the same title: http://www.expmath.org/ )

It seems I did some work on this too, as an experimental prelude for
the understanding of geometric Galois groups but nothing worth
publishing.

Olivier

[thargle...@hotmail.com]

You might like to take a look at the transparencies of a talk by
Benedicte Dujardin, on distribution of roots of random polynomials. It
has some good pictures.
See http://www-sop.inria.fr/miaou/anap03/bd-transparents.pdf
The full article appears in a book, soon to be published. See
http://www.springer.com/sgw/cda/frontpage/0,11855,3-40109-22-109680128-0,00.html

ZTW