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Density of Polynomial Roots

This investigation was inspired by the image at the top of the Experimental Mathematics Website under David H. Bailey's homepage at LBNL. At this point, I have reproduced some of the simpler results produced by Loki Jorgenson, see his Mathematics Unlimited Images and Zeros of Polynomials with Constrained Coefficients . The images below show polynomial root density with color and not the root sensitivity shown in Loki Jorgenson's images.

The data for the images below and the original images were generated using Matlab.


All of these images show the roots of families of polynomials colored by logarithm of their density. In these images, red/brown represents high density and blue represents low density. Note that there is a very high density at zero and along other portions of the real line. In the images, all these densities have been clipped/clamped so that the densities in the remainder of each image have the wider color range. Thus the colors along the real line are not representative of the densities there. In all the images the domain extends from -2-2i to 2+2i.


Density of roots of polynomials with coefficients in {-1,0,1} up to degree 16.
The domain extends from -2-2i to 2+2i.
(higher resolution image)


Density of roots of polynomials with coefficients in {-1,1} up to degree 21.
The domain extends from -2-2i to 2+2i.
(higher resolution image)

Density of roots of polynomials with coefficients in {-2,-1,0,1,2} up to degree 9.
The domain extends from -2-2i to 2+2i.
(higher resolution image)

Uncropped image where domain extends from -3-3i to 3+3i.


Density of roots of polynomials with coefficients in {0,1} up to degree 24.
The domain extends from -2-2i to 2+2i.
(higher resolution image)

New Results


Density of roots of polynomials with coefefficients in {-3,-2,-1,0,1,2,3} up to degree 8 (higher resolution image).

Uncropped image where domain extends from -3-3i to 3+3i.


Density of roots of polynomials with coefefficients in {-50,-49,...,0,...,49,50} up to degree 2 (higher resolution image).

Uncropped image where domain extends from -3-3i to 3+3i.

Density of roots of polynomials with coefefficients in {0,1,2} up to degree 14 (higher resolution image).


Density of roots of polynomials with coefefficients in {0,1,...,12} up to degree 5 (higher resolution image).

Uncropped image where domain extends from -3-3i to 3+3i.

Density of roots of polynomials with coefefficients in {0,1,...,30} up to degree 4 (higher resolution image).

Uncropped image where domain extends from -3-3i to 3+3i.


Density of roots of polynomials with coefefficients in {0,1,...,100} up to degree 3 (higher resolution image).

Uncropped image where domain extends from -3-3i to 3+3i.