Complex Zeros

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Complex Zeros

Polynomials with constrained coefficients have been much studied [1,14,4]. They relate to the Littlewood conjecture and many other problems. Littlewood notes [13] that ``these raise fascinating questions''.

Certain of these polynomials demonstrate amazing complexity when their zeros are appropriately plotted. Figure 11 shows the complex zeros of all polynomials

$\begin{displaymath} P_n(z) = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n\end{displaymath}$

of degree $n \le 18$ where $a_i = \{-1,+1\}$ as they appear on the complex plane. This image, reminiscent of pictures for polynomials with all coefficients in the set $\{0,+1\}$ [14], does raise many questions:

Is the set fractal and what is its boundary? Are there holes at infinite degree? How do the holes vary with the degree? What is the relationship between these zeros and those of polnomials with real coefficients in the neighbourhood of $\{-1,+1\}$ ?

Some, but definitely not all, of these questions have found some analytic answer [14,4]. However the images themselves offer a definitive description the polynomials. It remains to be seen how they will contribute to the general understanding of the visualization of numbers .

**Figure 11:** Roots of Littlewood Polynomials of degree at most 18 for coefficients $\pm 1$ . At left is the complete distribution of zeros on the complex plane. On the right are only the positive imaginary contributions colored by their sensitivity to variation in the polynomial coefficients.
$\begin{figure} \centerline{ \psfig {figure=FIGPOLY1-1BW.ps,width=2in} }\end{figure}$ $\begin{figure} \centerline{ \psfig {figure=FIGPOLY1-1BW.ps,width=2in} }\end{figure}$

Next: Conclusion Up: The Structure of Numbers Previous: Quasi-Rationals

loki@cecm.sfu.ca