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The computations of zeros graphed in our figures were performed in double precision (approx. 18 decimal places) on a Silicon Graphics workstation. Some of the zeros were checked for accuracy by recomputing them in double precision (approx. 28 decimal places) on a Cray X-MP. The zero-finding program used the Jenkins-Traub algorithm and was taken from a standard subroutine library. Checks showed that the values that were obtained were accurate on average to at least 10 decimal places, which was sufficient for our graphs. The program that was used appeared to produce accurate values on the Cray for the zeros for polynomials of degrees up to about 150. (Computation of zeros of polynomials of much higher degree would have required better algorithms, cf. [9].) Zeros of a large set of random polynomials of degree 100 were computed on the Cray, and they exhibit most of the features visible in Figures 1, 2 and 3. However, they are not as interesting as the lower degree zeros that are exhibited in Figures 1, 2 and 3. The ``spikes'' or ``tendrils'' that generate the fractal appearance in the graphs we include come from a small fraction of the polynomials. Sampling even of the polynomials of degree 100 does not yield a good representation of the extremal features that we expect to see for high as well as low degrees.

Graphs were prepared using the S system [2].

The graphs in
Figures 4,
5 and
6
were prepared differently.
A program was written that checked whether a given **w** with
**|w| < 1** is in .
Note that

The computations of
Figures 4,
5 and
6
are not completely rigorous in
that the determination of is rigorous,
while that of is not.
Moreover, an implicit premise in the preparation of
Figures 4,
5 and
6
was that if a point , then the whole
neighborhood of **w** represented by the corresponding pixel is in
.
On the other hand, the computations of
Figures 1,
2
and
3 are
rigorous.

It is possible to use computations to obtain rigorous estimates for that are sharper than those of Theorem 2.1. As an example, we sketch how a moderate amount of straightforward computing establishes that there are no with . We modify the method of proof of Theorem 2.1. Write

where Then we can write If we establish that on some simple closed contour about the origin, then by Rouché's theorem and will have the same number of zeros inside that contour. To prove that on a contourThe estimates used above were crude, and with more care one can either decrease the amount of computing (and even eliminate it altogether) or obtain better bounds for .

The basic principle that makes it possible to obtain good
estimates of is that for extremal points , the power series with
**0,1** coefficients such that are restricted.
For example the region depicted in
Figures 3 and
4 is

It should be possible to prove rigorously, with the help of numerical computations, such as those mentioned above, that the hole in mentioned in the Introduction and pictured in
Figure 6
is isolated in the
sense that there is a continuous closed curve in , for **U** a small rectangle, that encloses the hole.
We have not done this.

To explain the fractal appearance of ,
suppose that , **|w| < 1**, and that
where

** Acknowledgements**.
The authors thank E. R. Rodemich for originally raising the
question of the distribution of zeros of **0,1** polynomials,
M. Sambandham for providing a copy of [15],
A. R. Wilks for help with graphics, and D. Zagier for informing
them of the work of T. Bousch [5,6].
Special thanks are due to David desJardins and Emanuel Knill
for permission to use their proofs of
Lemma 5.1.

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