Mathematics Unlimited Images
Caption for cover image

The cover depicts the striking fractal pattern formed by plotting the complex zeros for all possible polynomials in powers of x with coefficients 1 and -1, up to degree 18. A wide variety of patterns and features become visible, leading researchers to new mathematical solutions. See page XXX for more details.


Figure 1a: Sensitivity of zeros for all polynomials with coefficients -1/1 up to degree 18 - Coloration is by sensitivity of the polynomials to slight variation around the values of the zeros. The color scale represents a normalized sensitivity to the range of values; red is insensitive to violet which is strongly sensitive. All zeros are pictured.

Figure 1b: Density of zeros for coefficients -1/1 - The zeros in figure 1a are colored by their local density. The range of color is normalized to the range of densities; red is low density to yellow which is high density.

Figure 1c: Sensitivity of zeros relative to coefficient a3 - Figure 1a is reproduced, this time with sensitivity relative to the x3 term. A subrange of the values is highlighted, with the least sensitive zeros outside the range colored in white.

Figure 1d: Sensitivity of zeros relative to coefficient a9 - Figure 1a is reproduced, this time with sensitivity relative to the x9 term. Note that the banded features appearing across the distribution of zeros are, to the best of knowledge, real and not programmatic artifacts. However, their mathematical explanation is not yet in hand.