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
where the 
are any 0,1 coefficients,
since we can write
The procedure was to test all sets of 0,1 coefficients
to see whether they could be the initial segment of coefficients of a power series
for which 
.
Let us regard the strings of coefficients 
as the
leaves of a balanced binary tree, with the nodes below the root corresponding to
, those below to 
, etc.
The procedure was to explore this tree,
checking whether
at any stage.
If 
is satisfied, then w is not a zero of any power series of
the form 
with initial coefficients
, and the subtree of that node does not have to be explored.
If all the leaves are discarded by this procedure, we have a rigorous
proof that 
, and so in fact an open neighborhood of w is outside 
.
On the other hand, if a leaf was found with
then the program assumed that 
.
(By establishing lower bounds for the derivative of the polynomial
at w and using crude upper bounds for
the second derivative, one could in principle prove that there is some point 
close to w such that 
, although the 10
in condition 
might have to be decreased.
Another way to prove this would be to use Lemma 3.1.
This step was not carried out.)
Figures 4,
5 and
6
were produced by testing each w in a 
or a 
grid (corresponding to the resolution of our laser
printer).
There were few points w for which neither condition
nor condition 
held.
The exceptions occur primarily in Fig. 4, but they do not affect
how the picture looks.
Had we used a tree of depth 80, the exceptions would have been
much more frequent.
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 contour C,
it suffices to show that 
on a discrete
set of points z on C, where
is such that bounds on the derivatives of 
and 
guarantee that 
will not decrease by
more than 
between the sampling points.
This was applied to each of the 
choices of
.
Of the 1024 functions 
, 997 satisfied 
on
The remaining 27 functions 
were shown to satisfy
on the contour
Finally, zeros of each of the 1024 polynomials 
were
computed, and it was found that 85 of these polynomials had a
single zero in 
, and the remaining 939 had none.
Thus in all cases we can conclude that 
has at most one
zero in 
.
Such a zero has to be real.
The 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
Numerical computation (evaluating polynomials of degrees 
with 0,1 coefficients at a 
uniform grid, and
bounding derivatives) shows that if 
,
then w can only be a zero of a power series of the form
This restricted the set of 
that had to be considered, and made
possible the computation of
Figure 3
, as it would not have been
feasible to examine all polynomials of degrees 
.
Furthermore, this restriction on the coefficients of 
makes it possible to estimate the shape of 
.
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
Suppose that
If 
and |z-w| is small, while d is large, we have
If 
(which as far as we know may hold for all
w with |w| < 1),
then 
for d large enough, and we can expect
that
Thus if
then we expect to find zeros in a neighborhood of each point of
The set 
is connected [1], and for 
,
it seems that it contains a small disk around the origin.
The set 
is a continuous function of w, which accounts
for the similarity of the protrusions
from 
visible in
Figures 5 and
6.
(The protrusions in Fig. 4 are different,
since there the sets 
are of different shape from those in
Figures 5 and
6.)
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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